Number system

2011 · Q35

A contract on construction job specifies a penalty for delay in completion of the work beyond a certain date is as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day etc., the penalty for each succeeding day being ₹ 50 more than that of the preceding day. How much penalty should the contractor pay if he delays the work by 10 days?

1. (a)₹ 4950
2. (b)₹ 4250UPSC key
3. (c)₹ 3600
4. (d)₹ 650

2011 · Q40

Three persons start walking together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

1. (a)25 m 20 cmUPSC key
2. (b)50 m 40 cm
3. (c)75 m 60 cm
4. (d)100 m 80 cm

2013 · Q28

There are five hobby clubs in a college viz., photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days?

1. (a)3UPSC key
2. (b)5
3. (c)10
4. (d)18

2013 · Q61

A gardener has 1000 plants. He wants to plant them in such a way that the number of rows and the number of columns remains the same. What is the minimum number of plants that he needs more for this purpose?

1. (a)14
2. (b)24UPSC key
3. (c)32
4. (d)34

2014 · Q6

Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 seconds respectively. The number of times they would fire the bullets together at the target in an hour is

1. (a)6
2. (b)7UPSC key
3. (c)8UPSC key
4. (d)9

2014 · Q22

A straight line segment is 36 cm long. Points are to be marked on the line from both the end points. From each end, the first point is at a distance of 1 cm from the end, the second point is at a distance of 2 cm from the first point and the third point is at a distance of 3 cm from the second point and so on. If the points on the ends are not counted and the common points are counted as one, what is the number of points?

1. (a)10
2. (b)12UPSC key
3. (c)14
4. (d)16

2014 · Q46

A bell rings every 18 minutes. A second bell rings every 24 minutes. A third bell rings every 32 minutes. If all the three bells ring at the same time at 8 o'clock in the morning, at what other time will they all ring together?

1. (a)12: 40 hrs
2. (b)12: 48 hrsUPSC key
3. (c)12: 56 hrs
4. (d)13: 04 hrs

2014 · Q77

For a charity show, the total tickets sold were 420. Half of these tickets were sold at the rate of ₹ 5 each, one-third at the rate of ₹ 3 each and the rest for ₹ 2 each. What was the total amount received?

1. (a)₹ 900
2. (b)₹ 1,540
3. (c)₹ 1,610UPSC key
4. (d)₹ 2,000

2015 · Q57

If ABC × DEED = ABCABC; where A, B, C, D and E are different digits, what are the values of D and E?

1. (a)D = 2, E = 0
2. (b)D = 0, E = 1
3. (c)D = 1, E = 0UPSC key
4. (d)D = 1, E = 2

2016 · Q13

There are five hobby clubs in a college — photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days?

1. (a)5
2. (b)18
3. (c)10
4. (d)3UPSC key

2016 · Q53

A person is standing on the first step from the bottom of a ladder. If he has to climb 4 more steps to reach exactly the middle step, how many steps does the ladder have?

1. (a)8
2. (b)9UPSC key
3. (c)10
4. (d)11

2016 · Q62

If R and S are different integers both divisible by 5, then which of the following is not necessarily true?

1. (a)R − S is divisible by 5
2. (b)R + S is divisible by 10UPSC key
3. (c)R × S is divisible by 25
4. (d)$R^{2}$ + $S^{2}$ is divisible by 5

2016 · Q63

How many numbers are there between 100 and 300 which either begin with or end with 2?

1. (a)110UPSC key
2. (b)111
3. (c)112
4. (d)None of the above

2016 · Q78

In a race, a competitor has to collect 6 apples which are kept in a straight line on a track and a bucket is placed at the beginning of the track which is a starting point. The condition is that the competitor can pick only one apple at a time, run back with it and drop it in the bucket. If he has to drop all the apples in the bucket, how much total distance he has to run if the bucket is 5 meters from the first apple and all other apples are placed 3 meters apart?

1. (a)40 m
2. (b)50 m
3. (c)75 m
4. (d)150 mUPSC key

2017 · Q12

If there is a policy that 1/3rd of a population of a community has migrated every year from one place to some other place, what is the leftover population of that community after the sixth year, if there is no further growth in the population during this period?

1. (a)16/243rd part of the population
2. (b)32/243rd part of the population
3. (c)32/729th part of the population
4. (d)64/729th part of the populationUPSC key

2017 · Q16

There are three pillars X, Y and Z of different heights. Three spiders A, B and C start to climb on these pillars simultaneously. In one chance, A climbs on X by 6 cm but slips down 1 cm. B climbs on Y by 7 cm but slips down 3 cm. C climbs on Z by 6·5 cm but slips down 2 cm. If each of them requires 40 chances to reach the top of the pillars, what is the height of the shortest pillar?

1. (a)161 cm
2. (b)163 cmUPSC key
3. (c)182 cm
4. (d)210 cm

2017 · Q33

A clock strikes once at 1 o'clock, twice at 2 o'clock and thrice at 3 o'clock, and so on. If it takes 12 seconds to strike at 5 o'clock, what is the time taken by it to strike at 10 o'clock?

1. (a)20 seconds
2. (b)24 secondsUPSC key
3. (c)28 seconds
4. (d)30 seconds

2017 · Q49

Certain 3-digit numbers have the following characteristics:

1.All the three digits are different.

2.The number is divisible by 7.

3.The number on reversing the digits is also divisible by 7.

How many such 3-digit numbers are there?

1. (a)2
2. (b)4UPSC key
3. (c)6
4. (d)8

2017 · Q51

How many numbers are there between 99 and 1000 such that the digit 8 occupies the units place?

1. (a)64
2. (b)80
3. (c)90UPSC key
4. (d)104

2017 · Q53

The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?

1. (a)42
2. (b)38UPSC key
3. (c)25
4. (d)16

2017 · Q73

A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder?

1. (a)9
2. (b)27
3. (c)36
4. (d)45UPSC key

2017 · Q79

There are certain 2-digit numbers. The difference between the number and the one obtained on reversing it is always 27. How many such maximum 2-digit numbers are there?

1. (a)3
2. (b)4
3. (c)5
4. (d)None of the aboveUPSC key

2017 · Q80

What is the total number of digits printed, if a book containing 150 pages is to be numbered from 1 to 150?

1. (a)262
2. (b)342UPSC key
3. (c)360
4. (d)450

2018 · Q2

Consider the following sum:

• + 1• + 2• + •3 + •1 = 21•

In the above sum, • stands for

1. (a)4
2. (b)5
3. (c)6
4. (d)8UPSC key

2018 · Q34

A number consists of three digits of which the middle one is zero and their sum is 4. If the number formed by interchanging the first and last digits is greater than the number itself by 198, then the difference between the first and last digits is

1. (a)1
2. (b)2UPSC key
3. (c)3
4. (d)4

2019 · Q9

The number of times the digit 5 will appear while writing the integers from 1 to 1000 is

1. (a)269
2. (b)271
3. (c)300UPSC key
4. (d)302

2019 · Q15

In a school every student is assigned a unique identification number. A student is a football player if and only if the identification number is divisible by 4, whereas a student is a cricketer if and only if the identification number is divisible by 6. If every number from 1 to 100 is assigned to a student, then how many of them play cricket as well as football?

1. (a)4
2. (b)8UPSC key
3. (c)10
4. (d)12

2019 · Q33

The ratio of a two-digit natural number to a number formed by reversing its digits is 4: 7. The number of such pairs is

1. (a)5
2. (b)4UPSC key
3. (c)3
4. (d)2

2019 · Q35

Seeta and Geeta go for a swim after a gap of every 2 days and every 3 days respectively. If on 1ˢᵗ January both of them went for a swim together, when will they go together next?

1. (a)7ᵗʰ January
2. (b)8ᵗʰ January
3. (c)12ᵗʰ January
4. (d)13ᵗʰ JanuaryUPSC key

2019 · Q50

If the numerator and denominator of a proper fraction are increased by the same positive quantity which is greater than zero, the resulting fraction is

1. (a)always less than the original fraction
2. (b)always greater than the original fractionUPSC key
3. (c)always equal to the original fraction
4. (d)such that nothing can be claimed definitely

2019 · Q53

A printer numbers the pages of a book starting with 1 and uses 3089 digits in all. How many pages does the book have?

1. (a)1040
2. (b)1048
3. (c)1049UPSC key
4. (d)1050

2019 · Q60

Number 136 is added to 5B7 and the sum obtained is 7A3, where A and B are integers. It is given that 7A3 is exactly divisible by 3. The only possible value of B is

1. (a)2
2. (b)5
3. (c)7
4. (d)8UPSC key

2019 · Q75

An 8-digit number 4252746B leaves remainder 0 when divided by 3. How many values of B are possible?

1. (a)2
2. (b)3
3. (c)4UPSC key
4. (d)6

2020 · Q7

How many zeroes are there at the end of the following product?

1 × 5 × 10 × 15 × 20 × 25 × 30 × 35 × 40 × 45 × 50 × 55 × 60

1. (a)10UPSC key
2. (b)12
3. (c)14
4. (d)15

2020 · Q8

Let XYZ be a three-digit number, where (X + Y + Z) is not a multiple of 3. Then (XYZ + YZX + ZXY) is not divisible by

1. (a)3
2. (b)9UPSC key
3. (c)37
4. (d)(X + Y + Z)

2020 · Q10

How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?

1. (a)ZeroUPSC key
2. (b)One
3. (c)Nine
4. (d)Ten

2020 · Q11

In the sum

⊗ + 1⊗ + 5⊗ + ⊗⊗ + ⊗1 = 1⊗⊗

for which digit does the symbol ⊗ stand?

1. (a)2
2. (b)3UPSC key
3. (c)4
4. (d)5

2020 · Q12

If you have two straight sticks of length 7·5 feet and 3·25 feet, what is the minimum length can you measure?

1. (a)0·05 foot
2. (b)0·25 footUPSC key
3. (c)1 foot
4. (d)3·25 feet

2020 · Q14

One page is torn from a booklet whose pages are numbered in the usual manner starting from the first page as 1. The sum of the numbers on the remaining pages is 195. The torn page contains which of the following numbers?

1. (a)5, 6
2. (b)7, 8UPSC key
3. (c)9, 10
4. (d)11, 12

2020 · Q16

Let A3BC and DE2F be four-digit numbers where each letter represents a different digit greater than 3. If the sum of the numbers is 15902, then what is the difference between the values of A and D?

1. (a)1
2. (b)2
3. (c)3UPSC key
4. (d)4

2020 · Q32

Two Statements are given followed by two Conclusions:

Statements:

All numbers are divisible by 2.

All numbers are divisible by 3.

Conclusion–I:

All numbers are divisible by 6.

Conclusion–II:

All numbers are divisible by 4.

Which of the above Conclusions logically follows/follow from the two given Statements?

1. (a)Only Conclusion–IUPSC key
2. (b)Only Conclusion–II
3. (c)Neither Conclusion–I nor Conclusion–II
4. (d)Both Conclusion–I and Conclusion–II

2020 · Q37

How many integers are there between 1 and 100 which have 4 as a digit but are not divisible by 4?

1. (a)5
2. (b)11
3. (c)12UPSC key
4. (d)13

2020 · Q51

What is the greatest length x such that 3 $\dfrac{1}{2}$ m and 8 $\dfrac{3}{4}$ m are integral multiples of x?

1. (a)1 $\dfrac{1}{2}$ m
2. (b)1 $\dfrac{1}{3}$ m
3. (c)1 $\dfrac{1}{4}$ m
4. (d)1 $\dfrac{3}{4}$ mUPSC key

2020 · Q53

The recurring decimal representation 1·272727... is equivalent to

1. (a)$\dfrac{13}{11}$
2. (b)$\dfrac{14}{11}$UPSC key
3. (c)$\dfrac{127}{99}$
4. (d)$\dfrac{137}{99}$

2020 · Q54

What is the least four-digit number when divided by 3, 4, 5 and 6 leaves a remainder 2 in each case?

1. (a)1012
2. (b)1022UPSC key
3. (c)1122
4. (d)1222

2020 · Q56

What is the remainder when 51 × 27 × 35 × 62 × 75 is divided by 100?

1. (a)50UPSC key
2. (b)25
3. (c)5
4. (d)1

2020 · Q58

For what value of n, the sum of digits in the number ($10^{n}$ + 1) is 2?

1. (a)For n = 0 only
2. (b)For any whole number nUPSC key
3. (c)For any positive integer n only
4. (d)For any real number n

2020 · Q75

How many pairs of natural numbers are there such that the difference of whose squares is 63?

1. (a)3UPSC key
2. (b)4
3. (c)5
4. (d)2

2020 · Q76

Which one of the following will have minimum change in its value if 5 is added to both numerator and the denominator of the fractions $\dfrac{2}{3}$, $\dfrac{3}{4}$, $\dfrac{4}{5}$ and $\dfrac{5}{6}$?

1. (a)$\dfrac{2}{3}$
2. (b)$\dfrac{3}{4}$
3. (c)$\dfrac{4}{5}$
4. (d)$\dfrac{5}{6}$UPSC key

2020 · Q77

A digit n > 3 is divisible by 3 but not divisible by 6. Which one of the following is divisible by 4?

1. (a)2n
2. (b)3n
3. (c)2n + 4
4. (d)3n + 1UPSC key

2020 · Q80

A frog tries to come out of a dried well 4·5 m deep with slippery walls. Every time the frog jumps 30 cm, slides down 15 cm. What is the number of jumps required for the frog to come out of the well?

1. (a)28
2. (b)29UPSC key
3. (c)30
4. (d)31

2021 · Q5

If $3^{2019}$ is divided by 10, then what is the remainder?

1. (a)1
2. (b)3
3. (c)7UPSC key
4. (d)9

2021 · Q6

The number 3798125P369 is divisible by 7. What is the value of the digit P?

1. (a)1
2. (b)6UPSC key
3. (c)7
4. (d)9

2021 · Q16

A boy plays with a ball and he drops it from a height of 1·5 m. Every time the ball hits the ground, it bounces back to attain a height $\dfrac{4}{5}$ᵗʰ of the previous height. The ball does not bounce further if the previous height is less than 50 cm. What is the number of times the ball hits the ground before the ball stops bouncing?

1. (a)4
2. (b)5UPSC key
3. (c)6
4. (d)7

2021 · Q19

Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10?

1. (a)6
2. (b)7
3. (c)8
4. (d)9UPSC key

2021 · Q36

Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.

Which of the following is/are correct?

1.S is always divisible by 74.

2.S is always divisible by 9.

Select the correct answer using the code given below:

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2UPSC key
4. (d)Neither 1 nor 2

2021 · Q59

Consider the following addition problem:

3P + 4P + PP + PP = RQ2; where P, Q and R are different digits.

What is the arithmetic mean of all such possible sums?

1. (a)102
2. (b)120
3. (c)202UPSC key
4. (d)220

2021 · Q60

Consider the following multiplication problem:

(PQ) × 3 = RQQ, where P, Q and R are different digits and R ≠ 0.

What is the value of (P + R) ÷ Q?

1. (a)1
2. (b)2UPSC key
3. (c)5
4. (d)Cannot be determined due to insufficient data

2021 · Q67

Consider the following statements:

1.The sum of 5 consecutive integers can be 100.

2.The product of three consecutive natural numbers can be equal to their sum.

Which of the above statements is/are correct?

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2UPSC key
4. (d)Neither 1 nor 2

2021 · Q75

Joseph visits the club on every 5ᵗʰ day, Harsh visits on every 24ᵗʰ day, while Sumit visits on every 9ᵗʰ day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?

1. (a)Monday
2. (b)WednesdayUPSC key
3. (c)Thursday
4. (d)Sunday

2021 · Q76

The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.

Consider the following statements:

1.The sum of the two digits of the number can be determined only if the product of the two digits is known.

2.The difference between the two digits of the number can be determined.

Which of the above statements is/are correct?

1. (a)1 only
2. (b)2 onlyUPSC key
3. (c)Both 1 and 2
4. (d)Neither 1 nor 2

2021 · Q79

When a certain number is multiplied by 7, the product entirely comprises ones only (1111...). What is the smallest such number?

1. (a)15713
2. (b)15723
3. (c)15783
4. (d)15873UPSC key

2022 · Q5

An Identity Card has the number ABCDEFG, not necessarily in that order, where each letter represents a distinct digit (1, 2, 4, 5, 7, 8, 9 only). The number is divisible by 9. After deleting the first digit from the right, the resulting number is divisible by 6. After deleting two digits from the right of original number, the resulting number is divisible by 5. After deleting three digits from the right of original number, the resulting number is divisible by 4. After deleting four digits from the right of original number, the resulting number is divisible by 3. After deleting five digits from the right of original number, the resulting number is divisible by 2. Which of the following is a possible value for the sum of the middle three digits of the number?

1. (a)8UPSC key
2. (b)9
3. (c)11
4. (d)12

2022 · Q10

The digits 1 to 9 are arranged in three rows in such a way that each row contains three digits, and the number formed in the second row is twice the number formed in the first row; and the number formed in the third row is thrice the number formed in the first row. Repetition of digits is not allowed. If only three of the four digits 2, 3, 7 and 9 are allowed to use in the first row, how many such combinations are possible to be arranged in the three rows?

1. (a)4
2. (b)3
3. (c)2UPSC key
4. (d)1

2022 · Q26

On one side of a 1·01 km long road, 101 plants are planted at equal distance from each other. What is the total distance between 5 consecutive plants?

1. (a)40 m
2. (b)40·4 mUPSC key
3. (c)50 m
4. (d)50·5 m

2022 · Q35

How many seconds in total are there in x weeks, x days, x hours, x minutes and x seconds?

1. (a)11580x
2. (b)11581x
3. (c)694860x
4. (d)694861xUPSC key

2022 · Q47

A person X wants to distribute some pens among six children A, B, C, D, E and F. Suppose A gets twice the number of pens received by B, three times that of C, four times that of D, five times that of E and six times that of F. What is the minimum number of pens X should buy so that the number of pens each one gets is an even number?

1. (a)147
2. (b)150
3. (c)294UPSC key
4. (d)300

2022 · Q54

Let A, B and C represent distinct non-zero digits. Suppose x is the sum of all possible 3-digit numbers formed by A, B and C without repetition.

Consider the following statements:

1.The 4-digit least value of x is 1332.

2.The 3-digit greatest value of x is 888.

Which of the above statements is/are correct?

1. (a)1 onlyUPSC key
2. (b)2 only
3. (c)Both 1 and 2
4. (d)Neither 1 nor 2

2022 · Q58

What is the remainder when

91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99

is divided by 1261?

1. (a)3
2. (b)2
3. (c)1
4. (d)0UPSC key

2022 · Q65

What is the smallest number greater than 1000 that when divided by any one of the numbers 6, 9, 12, 15, 18 leaves a remainder of 3?

1. (a)1063
2. (b)1073
3. (c)1083UPSC key
4. (d)1183

2022 · Q66

Let p be a two-digit number and q be the number consisting of same digits written in reverse order. If p × q = 2430, then what is the difference between p and q?

1. (a)45
2. (b)27
3. (c)18
4. (d)9UPSC key

2022 · Q67

Consider the following statements in respect of two natural numbers p and q such that p is a prime number and q is a composite number:

1.p × q can be an odd number.

2.q/p can be a prime number.

3.p + q can be a prime number.

Which of the above statements are correct?

1. (a)1 and 2 only
2. (b)2 and 3 only
3. (c)1 and 3 only
4. (d)1, 2 and 3UPSC key

2022 · Q74

If

15 × 14 × 13 × ⋯ × 3 × 2 × 1 = 3ᵐ × n

where m and n are positive integers, then what is the maximum value of m?

1. (a)7
2. (b)6UPSC key
3. (c)5
4. (d)4

2022 · Q78

The sum of three consecutive integers is equal to their product. How many such possibilities are there?

1. (a)Only one
2. (b)Only two
3. (c)Only threeUPSC key
4. (d)No such possibility is there

2023 · Q7

What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100?

1. (a)0UPSC key
2. (b)1
3. (c)2
4. (d)4

2023 · Q8

What is the unit digit in the expansion of $57242^{9\times 7\times 5\times 3\times 1}$?

1. (a)2UPSC key
2. (b)4
3. (c)6
4. (d)8

2023 · Q9

If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F?

1. (a)28
2. (b)29
3. (c)30
4. (d)31UPSC key

2023 · Q10

D is a 3-digit number such that the ratio of the number to the sum of its digits is least. What is the difference between the digit at the hundred's place and the digit at the unit's place of D?

1. (a)0
2. (b)7
3. (c)8UPSC key
4. (d)9

2023 · Q14

Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in order). Consider the following:

1.p + q + r − s − t is definitely even.

2.2p + q + 2r − 2s + t is definitely odd.

Which of the above statements is/are correct?

1. (a)1 onlyUPSC key
2. (b)2 only
3. (c)Both 1 and 2
4. (d)Neither 1 nor 2

2023 · Q15

Consider the following in respect of prime number p and composite number c.

1.(p + c)/(p − c) can be even.

2.2p + c can be odd.

3.pc can be odd.

Which of the statements given above are correct?

1. (a)1 and 2 only
2. (b)2 and 3 only
3. (c)1 and 3 only
4. (d)1, 2 and 3UPSC key

2023 · Q16

A 3-digit number ABC, on multiplication with D gives 37DD where A, B, C and D are different non-zero digits. What is the value of A + B + C?

1. (a)18UPSC key
2. (b)16
3. (c)15
4. (d)Cannot be determined due to insufficient data

2023 · Q17

For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by:

1. (a)7 and 11 only
2. (b)11 and 13 only
3. (c)7 and 13 only
4. (d)7, 11 and 13UPSC key

2023 · Q25

Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible?

1. (a)10
2. (b)11
3. (c)12UPSC key
4. (d)Infinitely many

2023 · Q26

If p, q, r and s are distinct single digit positive numbers, then what is the greatest value of (p + q)(r + s)?

1. (a)230
2. (b)225UPSC key
3. (c)224
4. (d)221

2023 · Q27

A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13?

1. (a)11UPSC key
2. (b)9
3. (c)7
4. (d)1

2023 · Q28

Each digit of a 9-digit number is 1. It is multiplied by itself. What is the sum of the digits of the resulting number?

1. (a)64
2. (b)80
3. (c)81UPSC key
4. (d)100

2023 · Q29

What is the sum of all digits which appear in all the integers from 10 to 100?

1. (a)855
2. (b)856UPSC key
3. (c)910
4. (d)911

2023 · Q45

How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers?

1. (a)6
2. (b)7
3. (c)8
4. (d)9UPSC key

2023 · Q46

Let pp, qq and rr be 2-digit numbers where p < q < r. If pp + qq + rr = tt0, where tt0 is a 3-digit number ending with zero, consider the following statements:

1.The number of possible values of p is 5.

2.The number of possible values of q is 6.

Which of the above statements is/are correct?

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2UPSC key
4. (d)Neither 1 nor 2

2023 · Q50

There are three traffic signals. Each signal changes colour from green to red and then from red to green. The first signal takes 25 seconds, the second signal takes 39 seconds and the third signal takes 60 seconds to change the colour from green to red. The durations for green and red colours are same. At 2:00 p.m, they together turn green. At what time will they change to green next, simultaneously?

1. (a)4:00 p.m.
2. (b)4:10 p.m.UPSC key
3. (c)4:20 p.m.
4. (d)4:30 p.m.

2023 · Q69

There are large number of silver coins weighing 2 gm, 5 gm, 10 gm, 25 gm, 50 gm each. Consider the following statements:

1.To buy 78 gm of coins one must buy at least 7 coins.

2.To weigh 78 gm using these coins one can use less than 7 coins.

Which of the statements given above is/are correct?

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2UPSC key
4. (d)Neither 1 nor 2

2023 · Q74

What is the remainder if $2^{192}$ is divided by 6?

1. (a)0
2. (b)1
3. (c)2
4. (d)4UPSC key

2023 · Q76

AB and CD are 2-digit numbers. Multiplying AB with CD results in a 3-digit number DEF. Adding DEF to another 3-digit number GHI results in 975. Further A, B, C, D, E, F, G, H, I are distinct digits. If E = 0, F = 8, then what is A + B + C equal to?

1. (a)6UPSC key
2. (b)7
3. (c)8
4. (d)9

2024 · Q6

In the expression 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1, ∗ is chosen from +, −, × each at most two times. What is the smallest non-negative value of the expression?

1. (a)3
2. (b)2
3. (c)1
4. (d)0UPSC key

2024 · Q9

How many consecutive zeros are there at the end of the integer obtained in the product $1^{2}$ × $2^{4}$ × $3^{6}$ × $4^{8}$ × ⋯ × $25^{50}$?

1. (a)50
2. (b)55
3. (c)100
4. (d)200UPSC key

2024 · Q10

On January 1st, 2023, a person saved ₹ 1. On January 2nd, 2023, he saved ₹ 2 more than that on the previous day. On January 3rd, 2023, he saved ₹ 2 more than that on the previous day and so on. At the end of which date was his total savings a perfect square as well a perfect cube?

1. (a)7th January, 2023
2. (b)8th January, 2023UPSC key
3. (c)9th January, 2023
4. (d)Not possible

2024 · Q15

$222^{333}$ + $333^{222}$ is divisible by which of the following numbers?

1. (a)2 and 3 but not 37
2. (b)3 and 37 but not 2UPSC key
3. (c)2 and 37 but not 3
4. (d)2, 3 and 37

2024 · Q17

What is the rightmost digit preceding the zeros in the value of $30^{30}$?

1. (a)1
2. (b)3
3. (c)7
4. (d)9UPSC key

2024 · Q18

421 and 427, when divided by the same number, leave the same remainder 1. How many numbers can be used as the divisor in order to get the same remainder 1?

1. (a)1
2. (b)2
3. (c)3UPSC key
4. (d)4

2024 · Q19

A can X contains 399 litres of petrol and a can Y contains 532 litres of diesel. They are to be bottled in bottles of equal size so that whole of petrol and diesel would be separately bottled. The bottle capacity in terms of litres is an integer. How many different bottle sizes are possible?

1. (a)3
2. (b)4UPSC key
3. (c)5
4. (d)6

2024 · Q20

Consider the following statements in respect of the sum S = x + y + z, where x, y and z are distinct prime numbers each less than 10:

1.The unit digit of S can be 0.

2.The unit digit of S can be 9.

3.The unit digit of S can be 5.

Which of the statements given above are correct?

1. (a)1 and 2 only
2. (b)2 and 3 only
3. (c)1 and 3 onlyUPSC key
4. (d)1, 2 and 3

2024 · Q35

Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, then what is the number of possible values of X?

1. (a)2
2. (b)4
3. (c)6
4. (d)8UPSC key

2024 · Q39

Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements:

1.$(p − r)^{2}$(qs) is even.

2.(q − s)$q^{2}$s is even.

3.$(q + r)^{2}$(p + s) is odd.

Which of the statements given above are correct?

1. (a)1 and 2 only
2. (b)2 and 3 only
3. (c)1 and 3 only
4. (d)1, 2 and 3UPSC key

2024 · Q45

What is the number of fives used in numbering a 260-page book?

1. (a)55
2. (b)56UPSC key
3. (c)57
4. (d)60

2024 · Q48

If the sum of the two-digit numbers AB and CD is the three-digit number 1CE, where the letters A, B, C, D, E denote distinct digits, then what is the value of A?

1. (a)9UPSC key
2. (b)8
3. (c)7
4. (d)Cannot be determined due to insufficient data

2024 · Q54

$32^{5}$ + $2^{27}$ is divisible by

1. (a)3
2. (b)7
3. (c)10UPSC key
4. (d)11

2024 · Q55

Let p and q be positive integers satisfying p < q and p + q = k. What is the smallest value of k that does not determine p and q uniquely?

1. (a)3
2. (b)4
3. (c)5UPSC key
4. (d)6

2025 · Q5

A natural number N is such that it can be expressed as N = p + q + r, where p, q and r are distinct factors of N. How many numbers below 50 have this property?

1. (a)6
2. (b)7
3. (c)8UPSC key
4. (d)9

2025 · Q6

Three prime numbers p, q and r, each less than 20, are such that p − q = q − r. How many distinct possible values can we get for (p + q + r)?

1. (a)4
2. (b)5
3. (c)6
4. (d)More than 6UPSC key

2025 · Q7

How many possible values of (p + q + r) are there satisfying $\dfrac{1}{p}$ + $\dfrac{1}{q}$ + $\dfrac{1}{r}$ = 1, where p, q and r are natural numbers (not necessarily distinct)?

1. (a)None
2. (b)One
3. (c)ThreeUPSC key
4. (d)More than three

2025 · Q18

A 4-digit number N is such that when divided by 3, 5, 6, 9 leaves a remainder 1, 3, 4, 7 respectively. What is the smallest value of N?

1. (a)1068
2. (b)1072
3. (c)1078UPSC key
4. (d)1082

2025 · Q20

What is the unit digit in the multiplication of 1 × 3 × 5 × 7 × 9 × ... × 999?

1. (a)1
2. (b)3
3. (c)5UPSC key
4. (d)9

2025 · Q25

Consider the first 100 natural numbers. How many of them are not divisible by any one of 2, 3, 5, 7 and 9?

1. (a)20
2. (b)21
3. (c)22UPSC key
4. (d)23

2025 · Q27

Let both p and k be prime numbers such that ($p^{2}$ + k) is also a prime number less than 30. What is the number of possible values of k?

1. (a)4
2. (b)5UPSC key
3. (c)6
4. (d)7

2025 · Q28

There are n sets of numbers each having only three positive integers with LCM equal to 1001 and HCF equal to 1. What is the value of n?

1. (a)6
2. (b)7
3. (c)8
4. (d)More than 8UPSC key

2025 · Q29

Let PQR be a 3-digit number, PPT be a 3-digit number and PS be a 2-digit number, where P, Q, R, S, T are distinct non-zero digits. Further, PQR − PS = PPT. If Q = 3 and T < 6, then what is the number of possible values of (R, S)?

1. (a)2
2. (b)3UPSC key
3. (c)4
4. (d)More than 4

2025 · Q35

What is the maximum value of n such that 7 × 343 × 385 × 1000 × 2401 × 77777 is divisible by $35^{n}$?

1. (a)3
2. (b)4UPSC key
3. (c)5
4. (d)7

2025 · Q40

If $N^{2}$ = 12345678987654321, then how many digits does the number N have?

1. (a)8
2. (b)9UPSC key
3. (c)10
4. (d)11

2025 · Q46

If n is a natural number, then what is the number of distinct remainders of ($1^{n}$ + $2^{n}$) when divided by 4?

1. (a)0
2. (b)1
3. (c)2UPSC key
4. (d)3

2025 · Q47

Let P = QQQ be a 3-digit number. What is the HCF of P and 481?

1. (a)1
2. (b)13
3. (c)37UPSC key
4. (d)481

2025 · Q48

What is the 489th digit in the number 123456789101112...?

1. (a)0
2. (b)3
3. (c)6
4. (d)9UPSC key

2025 · Q64

Consider the following statements:

I.There exists a natural number which when increased by 50% can have its number of factors unchanged.

II.There exists a natural number which when increased by 150% can have its number of factors unchanged.

Which of the statements given above is/are correct?

1. (a)I only
2. (b)II only
3. (c)Both I and IIUPSC key
4. (d)Neither I nor II

2025 · Q67

What is the remainder when $9^{3}$ + $9^{4}$ + $9^{5}$ + $9^{6}$ + ... + $9^{100}$ is divided by 6?

1. (a)0UPSC key
2. (b)1
3. (c)2
4. (d)3

2025 · Q80

The difference between any two natural numbers is 10. What can be said about the natural numbers which are divisible by 5 and lie between these two numbers?

1. (a)There is only one such number.
2. (b)There are only two such numbers.
3. (c)There can be more than one such number.UPSC key
4. (d)No such number exists.

2026 · Q10

X, Y and Z jump forward 4′, 6′ and 5′, respectively. At 8 AM, they all land on mark 199′. How many times will they all land on the same mark (need not be at the same moment) between mark 195′ and 1000′, if all of them cross mark 1000′ by 9 AM?

1. (a)11
2. (b)12
3. (c)13
4. (d)14UPSC key

2026 · Q19

If 10ᵐ × 1000 × n = $75^{25}$ × $25^{32}$ × $32^{75}$, where n is not divisible by 10, then the value of m is

1. (a)101
2. (b)111UPSC key
3. (c)121
4. (d)131

2026 · Q22

What is the minimum number of times one needs to measure to get 298 litres of water from a tank, if the measuring cylinders have capacities 1 litre, 6 litres, 25 litres and 100 litres?

1. (a)4
2. (b)5UPSC key
3. (c)9
4. (d)13

2026 · Q29

Three variables x, y and z take values 2, 3, 4 or 5 such that their values are always distinct. If M and N denote the largest possible value and the smallest possible value, respectively, for the expression {(x × y) + z}; then M − N is

1. (a)11
2. (b)12
3. (c)13UPSC key
4. (d)14

2026 · Q42

The digit in the unit place of the number $6^{129}$ × $7^{307}$ is

1. (a)2
2. (b)4
3. (c)8UPSC key
4. (d)6

2026 · Q45

How many three-digit numbers can be expressed as an integral power of 2?

1. (a)1
2. (b)2
3. (c)3UPSC key
4. (d)4

2026 · Q49

How many times does 5 appear in all two-digit positive integers?

1. (a)18
2. (b)19UPSC key
3. (c)20
4. (d)21

2026 · Q59

If the product of the HCF and LCM of two distinct numbers is the cube of one of the numbers, then which of the following statements is/are correct?

I.The difference of the numbers is an even number.

II.One of the numbers is a perfect square.

Select the answer using the code given below.

1. (a)I only
2. (b)II only
3. (c)Both I and IIUPSC key
4. (d)Neither I nor II

2026 · Q60

If x and y are two digits and the number 4x5y790 is divisible by 11, then what is the remainder, if x + y is divided by 11?

1. (a)1
2. (b)3
3. (c)5
4. (d)7UPSC key

2026 · Q66

A is a 2-digit number with different digits. B is also a 2-digit number and is obtained by reversing the digits of A. If A − B is a multiple of 27, where A > B, how many such different A’s are possible?

1. (a)6
2. (b)9UPSC key
3. (c)12
4. (d)18