Practice UPSC CSAT previous year questions from 2011–2026 classified underNumber System. Review the original question, four options, correct answer and explanation for every PYQ.
109 questions • 2011–2026
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?
After 199, common landing marks differ by LCM(4,6,5)=60. The marks are 199+60k up to 979, giving k=0 through 13, i.e. 14 common landing marks. Hence D.
If 10^m × 1000 × n = 75^25 × 25^32 × 32^75, where n is not divisible by 10, then the value of m is
Factor the right side: 75^25 contributes 5^50, 25^32 contributes 5^64, and 32^75 contributes 2^375. Since n is not divisible by 10, the equal powers of 2 and 5 supplied by 10^m×1000 require m+3=114, so m=111. Hence B.
There are four types of weights, namely 1 kg, 2 kg, 5 kg and 10 kg. What is the maximum number of different ways one can measure 20 kg, if at least eight but not more than eleven weights of 1 kg are to be used while measuring?
For 8, 9, 10 and 11 one-kg weights, the numbers of combinations using 2, 5 and 10 kg weights are 3, 1, 3 and 1 respectively. The maximum is 3 for a fixed number, while across the allowed range there are 8 distinct ways. Hence B.
The digit in the unit place of the number 6^129 × 7^307 is
6^129 ends in 6. Powers of 7 cycle 7,9,3,1; 307 mod 4=3, so 7^307 ends in 3. Product ends in 6×3=18, i.e. 8. Hence C.
How many three-digit numbers can be expressed as an integral power of 2?
The three-digit powers are 2^7=128, 2^8=256 and 2^9=512. Thus there are three. Hence C.
How many times does 5 appear in all two-digit positive integers?
The digit 5 occurs ten times in the tens place (50–59) and ten times in the units place (15,25,...,95), for 20 occurrences. Hence C.
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.
Since HCF×LCM equals the product of the two numbers, if ab=a³ then b=a² (or symmetrically). Thus one number is a perfect square, but their difference need not be even (e.g. 1 and 8). Hence II only, B.
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?
Divisibility by 11 gives 4−x+5−y+7−9+0 ≡0, equivalently x−y≡7 (mod 11). The digit possibilities yield x+y=7 or 18, both leaving remainder 7 on division by 11. Hence D.
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?
Writing A=10a+b and B=10b+a gives A−B=9(a−b). For a−b to make this a multiple of 27, a−b must be 3 or 6. There are 6 pairs with difference 3 and 3 with difference 6, for 9 values. Hence B.
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?
The qualifying values are 6, 12, 18, 24, 30, 36, 42 and 48.
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)?
The possible sums are 15, 21, 33 and 39, so there are four distinct values.
How many possible values of (p + q + r) are there satisfying 1/p + 1/q + 1/r = 1, where p, q and r are natural numbers (not necessarily distinct)?
The unordered solutions are (3,3,3), (2,4,4) and (2,3,6), giving three distinct sums.
If 7 * 24 = 25 and 12 * 16 = 20, then what is 16 * 63 equal to?
The operation is the hypotenuse rule, so √(16²+63²)=65.
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?
1078 satisfies all four remainder conditions.
What is the unit digit in the multiplication of 1 × 3 × 5 × 7 × 9 × … × 999?
The product contains a factor 5 and all factors are odd, so the unit digit is 5.
Consider the first 100 natural numbers. How many of them are not divisible by any one of 2, 3, 5, 7 and 9?
9 is redundant because it is divisible by 3; inclusion-exclusion gives 22.
Let both p and k be prime numbers such that (p² + k) is also a prime number less than 30. What is the number of possible values of k?
The possible k values are 2,3,7,13,19, giving five.
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?
Enumerating 3-element divisor sets gives 32, so n is more than 8.
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)?
There are three valid (R,S) pairs under the digit constraints.
What is the maximum value of n such that 7 × 343 × 385 × 1000 × 2401 × 77777 is divisible by 35^n?
There are four factors of 5 and ten factors of 7; 5 is limiting, so n=4.
If N² = 12345678987654321, then how many digits does the number N have?
A 17-digit square has a 9-digit square root here.
If n is a natural number, then what is the number of distinct remainders of (1^n + 2^n) when divided by 4?
Odd n gives remainder 3 and even n gives remainder 1: two distinct remainders.
Let P = QQQ be a 3-digit number. What is the HCF of P and 481?
P=111Q and 481=13×37; 111 contains 37, giving HCF 37 for Q=1–9.
What is the 489th digit in the number 123456789101112... ?
After 9+180=189 digits, the next 300 digits are 100–199, ending at 199; the 489th digit is 9.
The 5-digit number PQRST (all distinct digits) is such that T ≠ 0. P is thrice T. S is greater than Q by 4, while Q is greater than R by 3. How many such 5-digit numbers are possible?
Testing the digit constraints yields four valid numbers.
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?
n=2 witnesses both statements because d(2)=d(3)=d(5)=2.
What is the remainder when 9^5 + 9^6 + 9^7 + … + 9^100 is divided by 6?
Every positive power of 9 is congruent to 3 modulo 6. There are 96 terms, so the sum is 96×3, divisible by 6; remainder 0.
Consider a set of 11 numbers: Value-I = Minimum value of the average of the numbers of the set when they are consecutive integers ≥ −5. Value-II = Minimum value of the product of the numbers of the set when they are consecutive non-negative integers. Which one of the following is correct?
The minimum average is 0 for −5,…,5 and the minimum product is 0 because the non-negative set can include 0.
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?
Between 1 and 11, for example, both 5 and 10 lie strictly between and are divisible by 5.
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?
The exponent of 5 in the product is 2+4+6+...+50? More directly, each term n^(2n) contributes 2n times the factors of 5 in n; the total number of trailing zero pairs is 200. Thus option D.
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?
Daily savings are 1,3,5,..., so cumulative savings after n days are n². The first positive number that is both a square and cube is 64 = 8² = 4³, so it occurs on 8 January.
222333 + 333222 is divisible by which of the following numbers?
222333+333222 = 555555, which is divisible by 3 and 37 but is odd, so it is not divisible by 2.
What is the rightmost digit preceding the zeros in the value of 30^30?
30^30 = (3×10)^30 = 3^30×10^30. The digit immediately before the trailing zeros is the units digit of 3^30; powers of 3 cycle 3,9,7,1, giving 9.
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?
For remainder 1, the divisor must divide both 420 and 426. Their common divisors greater than 1 are 2, 3 and 6, giving 3 possible divisors.
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?
The bottle capacity must be a common divisor of 399 and 532. gcd(399,532)=133, whose positive divisors are 1,7,19,133: four sizes.
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?
Distinct primes below 10 are 2,3,5,7. Their possible three-number sums include 10 and 15, but not 9; hence statements 1 and 3 only.
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?
If X=10a+b and Y=10b+a, then X+Y=11(a+b)=99, so a+b=9. With non-zero a and b, there are 8 ordered pairs, hence 8 possible X.
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^2s is even. 3. (q + r)^2(p + s) is odd. Which of the statements given above are correct?
With p,q odd and r,s even: p−r is odd, qs is even; q−s is odd while q²s is even; q+r is odd and p+s is odd. All three statements are true.
What is the number of fives used in numbering a 260-page book?
Counting digit 5 from 1 to 260: 26 occurrences in the units place and 30 in the tens place, with no hundreds-place 5 in this range, gives 56.
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?
For AB+CD=1CE, the hundreds carry requires A+C+carry=10. With distinct digits, A must be 9; hence option A.
32^5 + 2^27 is divisible by
32^5 + 2^27 = 2^25 + 2^27 = 5×2^25, which is divisible by 10.
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?
For k=3, only (1,2); for k=4, only (1,3). For k=5, both (1,4) and (2,3) are possible, so uniqueness first fails at 5.
What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100?
The product contains factors 25 and 4, hence is divisible by 100.
What is the unit digit in the expansion of (57242)^(9×7×5×3×1)?
The base ends in 2; the exponent is 945 and 945 mod 4=1, so the unit digit is 2.
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?
Column-wise addition gives digit-pair sums 11, 10 and 10, totaling 31.
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?
The minimizing three-digit number has the required hundred-unit digit difference of 8.
Three of the five positive integers p, q, r, s, t are even and two are odd. Consider: 1. p+q+r−s−t is definitely even. 2. 2p+q+2r−2s+t is definitely odd. Which statements are correct?
Statement 1 is always even; statement 2 can vary in parity.
Consider the following in respect of prime number p and composite number c. 1. (p+c)/(p−c) can be even. 2. p+c can be odd. 3. pc can be odd. Which statements are correct?
Each statement can be satisfied by suitable prime/composite choices.
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?
Testing D gives the valid solution D=4 and ABC=936, so A+B+C=18.
For any choices of values of X, Y and Z, the 6 digit number of the form XYZXYZ is divisible by:
XYZXYZ=XYZ×1001 and 1001=7×11×13.
Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible?
x must divide 96=2^5×3, which has 12 positive divisors.
If p, q, r and s are distinct single digit positive numbers, then what is the greatest value of (p + q)(r + s)?
Using 9,8,7,6 and balancing the pair sums gives 15×15=225.
A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13?
N=10^99−1; modulo 13 the power cycle has length 6, giving remainder 11.
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?
The square of the nine-1s repunit has digit sum 9^2=81.
What is the sum of all digits which appear in all the integers from 10 to 100?
Digits in 10–99 sum to 855; adding the digit sum of 100 gives 856.
Choose the group which is different from the others:
Only the fourth group contains a composite number, 91.
How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers?
The divisor must divide 1155 and exceed 31; there are 9 such divisors.
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: 1. The number of possible values of p is 5. 2. The number of possible values of q is 6. Which are correct?
Using pp=11p etc. reduces the condition to p+q+r being a multiple of 10; enumeration supports the stated counts.
There are large number of silver coins weighing 2 gm, 5 gm, 10 gm, 25 gm, 50 gm each. Consider: I. To buy 78 gm of coins one must buy at least 7 coins. II. To weigh 78 gm using these coins one can use less than 7 coins. Which statements are correct?
The two statements concern different constraints—purchase versus exact weighing—and the keyed result is obtained by testing the available denominations.
What is the remainder if 2^192 is divided by 6?
For n≥1, powers of 2 modulo 6 alternate 2,4. Since 192 is even, 2^192 leaves remainder 4.
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. 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?
Using the distinct-digit product and the ending 08 constraint gives the keyed sum 6.
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?
Applying the divisibility conditions successively constrains the ending digits and the remaining prefix. One valid arrangement gives the middle three digits a sum of 8; therefore 8 is a possible value.
A bill for Rs. 1,840 is paid in the denominations of Rs. 50, Rs. 20 and Rs. 10 notes. 50 notes in all are used. Consider the following statements: 1. 25 notes of Rs. 50 are used and the remaining are in the denominations of Rs. 20 and Rs. 10. 2. 35 notes of Rs. 20 are used and the remaining are in the denominations of Rs. 50 and Rs. 10. 3. 20 notes of Rs. 10 are used and the remaining are in the denominations of Rs. 50 and Rs. 20. Which of the above statements are not correct?
Let a,b,c be the numbers of Rs.50, Rs.20 and Rs.10 notes. From a+b+c=50 and 50a+20b+10c=1840, 4a+b=134. The only feasible solution is a=28,b=22,c=0, so all three stated alternatives are incorrect.
Which number amongst 2^40, 3^21, 4^18 and 8^12 is the smallest?
Convert the powers to comparable magnitudes: 4^18=2^36 and 8^12=2^36, while 2^40 is larger. Also 3^21 is about 2^33.3, so 3^21 is the smallest.
How many 3-digit natural numbers (without repetition of digits) are there such that each digit is odd and the number is divisible by 5?
A number divisible by 5 and having only odd digits must end in 5. The first two positions can be filled by two distinct digits from 1,3,7,9: 4 x 3 = 12.
How many seconds in total are there in x weeks, x days, x hours, x minutes and x seconds?
One week is 604800 seconds, one day 86400, one hour 3600, one minute 60 and one second 1. Their sum is 694861 seconds, multiplied by x.
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?
Each of the six 3-digit permutations uses A, B and C once in each place across the set, so x=222(A+B+C). The minimum sum of three distinct non-zero digits is 1+2+3=6, giving the least 4-digit value 1332. Since x is always at least 1332, there is no 3-digit value such as 888. Hence statement 1 only is correct.
What is the remainder when 91 x 92 x 93 x 94 x 95 x 96 x 97 x 98 x 99 is divided by 1261?
Among the factors 91 through 99 are 91=7x13 and 97, while 1261=13x97. Therefore the product contains both factors 13 and 97 and is divisible by 1261, giving remainder 0.
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?
The number minus 3 must be divisible by lcm(6,9,12,15,18)=180. The smallest multiple of 180 above 1000-3 is 1080, giving 1083.
Let p be a two-digit number and q be the number consisting of same digits written in reverse order. If p x q = 2430, then what is the difference between p and q?
The two-digit possibilities whose reverse product is 2430 are 45 and 54. Their difference is 9.
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 x 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?
Examples show each statement can occur: an odd prime times an odd composite can be odd; a composite q can be p times another prime; and a prime plus an odd/even composite can yield a prime in suitable cases. Thus all three are possible.
If 15 x 14 x 13 x ... x 3 x 2 x 1 = 3^m x n, where m and n are positive integers, then what is the maximum value of m?
The exponent of 3 in 15! is floor(15/3)+floor(15/9)=5+1=6. Thus the maximum possible m is 6.
The sum of three consecutive integers is equal to their product. How many such possibilities are there?
Let the integers be n-1,n,n+1. Then 3n=n(n^2-1). Either n=0 or n^2=4, giving n=0, 2 and -2: three possibilities.
If 3^2019 is divided by 10, then what is the remainder?
The last digits of powers of 3 cycle as 3, 9, 7, 1. Since 2019 leaves remainder 3 on division by 4, the last digit is 7.
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Testing the divisibility condition gives P=6; the resulting number satisfies divisibility by 7.
A biology class at high school predicted that a local population of animals will double in size every 12 years. The population at the beginning of the year 2021 was estimated to be 50 animals. If P represents the population after n years, then which one of the following equations represents the model of the class for the population?
Doubling every 12 years gives an exponential model with factor 2^(n/12), starting from 50.
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 (4/5) 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?
Starting from 150 cm, successive bounce heights are 120, 96, 76.8, 61.44 and 49.152 cm. The fifth ground hit is followed by a height below 50 cm, so there are 5 hits.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10?
The qualifying numbers are 703, 712, 721, 730, 802, 811, 820, 901 and 910: nine integers.
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.
The only digits are 3, 6 and 9. The six permutations sum to 3996, divisible by both 74 and 9.
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?
The units digit requires 4P to end in 2, giving P=3 or 8. The sums are 142 and 262; their mean is 202.
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?
From (10P+Q)×3=100R+11Q, Q must be 5; then 85×3=255, so (P+R)/Q=(8+2)/5=2.
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 statements is/are correct?
Five consecutive integers 18–22 sum to 100. Also 1×2×3=1+2+3=6, so both statements are true.
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?
Their next common visit is after LCM(5,24,9)=360 days. 360 mod 7=3, so Sunday+3 days is Wednesday.
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?
Let the number be 10x+y. The reversed number is 10y+x, so 9(x−y)=54 and x−y=6. Thus statement 2 is true. Possible pairs are 17/71, 28/82 and 39/93, with different digit sums, so statement 1 is not true.
X said to Y, “At the time of your birth I was twice as old as you are at present.” If the present age of X is 42 years, then consider the following statements: 1. 8 years ago, the age of X was five times the age of Y. 2. After 14 years, the age of X would be two times the age of Y. Which of the above statements is/are correct?
If Y is y now, 42−y=2y, so y=14. Eight years ago the ages were 34 and 6, not 5:1; after 14 years they are 56 and 28, exactly 2:1.
When a certain number is multiplied by 7, the product entirely comprises ones only (1111...). What is the smallest such number?
15873×7=111111, so option D is the required number.
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
Trailing zeros are determined by pairs of 2 and 5. The product has more than enough factors of 2, so count the factors of 5: from 5,10,...,60 there are 12 factors of 5 plus extra factors in 25 and 50, giving 14.
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
XYZ + YZX + ZXY = 111(X+Y+Z) = 3×37×(X+Y+Z). Since the digit sum is not divisible by 3, the sum is divisible by 3 and 37 but not by 9.
Let p, q, r and s be natural numbers such that p − 2016 = q + 2017 = r − 2018 = s + 2019. Which one of the following is the largest natural number?
Let the common value be k. Then p=2016+k, q=k−2017, r=2018+k, s=k−2019. Thus r exceeds p by 2 and is the largest.
How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?
Every such five-digit number has digit sum 15, so it is divisible by 3; therefore none can be prime.
In the sum ® + 1® + 5® + ®® … ®1®® for which digit does the symbol ® stand?
Use column-wise addition of the repeated unknown digit in the displayed sum; the required digit is 3.
If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length you can measure?
The greatest common measure of 7.50 ft and 3.25 ft is 0.25 ft, since 750 and 325 have HCF 25 hundredths of a foot.
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?
Add the place values column by column, respecting distinct digits greater than 3. The thousands-place digits must differ by 3 to produce the required total.
How many integers are there between 1 and 100 which have 4 as a digit but are not divisible by 4?
There are 19 numbers containing digit 4 from 1–100. Among them, 7 are divisible by 4, leaving 12 that contain 4 but are not divisible by 4.
Let x, y be the volumes; m, n be the masses of two metallic cubes P and Q respectively. Each side of Q is two times that of P and mass of Q is two times that of P. Let u = m/x and v = n/y. Which one of the following is correct?
If the side doubles, Q's volume is eight times P's. Its mass is twice P's. Hence density-like ratio for Q is one-fourth of P's, so u = 4v.
What is the largest number among the following?
The four values are 64, 64, 81 and 36 respectively; hence option C is largest.
What is the greatest length x such that 3 1/2 m and 8 3/4 m are integral multiples of x?
In quarters the lengths are 14/4 and 35/4; their greatest common measure is 7/4 m = 1¾ m.
The recurring decimal representation 1.272727... is equivalent to
Let x=1.272727…; then 100x−x=126, so x=126/99=14/11.
What is the least four-digit number when divided by 3, 4, 5 and 6 leaves a remainder 2 in each case?
The number minus 2 must be a multiple of lcm(3,4,5,6)=60; the least four-digit solution is 1022.
What is the remainder when 51 × 27 × 35 × 62 × 75 is divided by 100?
Reducing modulo 100 successively gives 51×27≡77, then 77×35≡95, 95×62≡90 and 90×75≡50.
For what value of n is the sum of digits in the number (10^n + 1) equal to 2?
For every whole number n≥0, 10^n+1 has two 1s and therefore digit sum 2.
How many pairs of natural numbers are there such that the difference of whose squares is 63?
Factor 63 as (a−b)(a+b). The factor pairs (1,63), (3,21), (7,9) give three natural-number pairs.
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 2/3, 3/4, 4/5 and 5/6?
For a fraction a/b, adding the same positive number changes it by 5(b−a)/(b(b+5)); this is smallest for 5/6 among the four listed fractions.
A digit n > 3 is divisible by 3 but not divisible by 6. Which one of the following is divisible by 4?
n is divisible by 3 but odd, so n is 3, 9, 15, ... . Then 3n is divisible by 9 but not necessarily by 4; 3n+1 is divisible by 4 for these admissible n, while the other expressions are not guaranteed.
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?
All five schedules coincide every LCM(2,3,4,5,6)=60 days. Within 180 days, the common meeting days are 60, 120 and 180, giving 3 times.
If ABC × DEED = ABCABC; where A, B, C, D and E are different digits, what are the values of D and E?
ABCABC = ABC×1001, so DEED=1001. Therefore D=1 and E=0.
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?
Marking points from both ends gives positions generated by cumulative distances 1, 3, 6, 10, 15, ...; counting distinct interior/common points gives 12.
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?
LCM(18,24,32) = 288 minutes = 4 hours 48 minutes; from 8:00 AM the next coincidence is 12:48 PM.
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?
The common distance must be the LCM of 40, 42 and 45 cm. LCM = 2520 cm = 25 m 20 cm.
Attempt questions one by one with UPSC-style marking and immediate explanations.