Number System & Divisibility Rising

The weight of X, in kg, is denoted by X. The weights of A, B, C, D, P, Q, R and S are measured. Given: A + B + C + D = 17 A + C = 6 P + Q + S + D = 15 P + Q + R + B = 17 P = R and Q = S Which one of the following stateme

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 cr

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

For \frac{1}{3} < x < y < 2, which of the following statements is/are always correct? I. x + \frac{1}{x} < y + \frac{1}{y} II. \frac{\sqrt{1+y^2}}{y} < \frac{\sqrt{1+x^2}}{x}

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 meas

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 xy + z, then

In an objective type question paper, 5 marks are awarded for a correct answer and 2 marks are deducted for a wrong answer. A student attempted all the questions and got a score of 69. Had he been awarded 4 marks for a co

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

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

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

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 numb

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?

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?

There are three types of rectangular tiles : 3' \times 3', 3' \times 7' and 3' \times 11'. An area of rectangular shape of dimensions 3' \times 100' is to be covered using these tiles without breaking them. If x and y ar

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 ?

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) ?

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) ?

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 ?

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

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

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 ?

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 ?

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 o

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

If N² = 12345678987654321, then how many digits does the number N have ?

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

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

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

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 ?

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 numb

What is the remainder when 9³ + 9⁴ + 9⁵ + 9⁶ + . . . + 9¹⁰⁰ is divided by 6 ?

Let p + q = 10, where p, q are integers. Value-I = Maximum value of p × q when p, q are positive integers. Value-II = Maximum value of p × q when p ≥ −6, q ≥ −4. Which one of the following is correct ?

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 co

Let x be a real number between 0 and 1. Which of the following statements is/are correct ? I. x² > x³. II. x > √x. Select the correct answer using the code given below :

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 ?

How many consecutive zeros are there at the end of the integer obtained in the product 1² × 2⁴ × 3⁶ × 4⁸ × ⋯ × 25⁵⁰?

222³³³ + 333²²² is divisible by which of the following numbers?

What is the rightmost digit preceding the zeros in the value of 30³⁰?

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?

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 term

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

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?

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)q²s is even. 3. (q + r)²(p + s) is odd. Which of the statements

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

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?

32⁵ + 2²⁷ is divisible by

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?

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

What is the unit digit in the expansion of (57242)^(9×7×5×3×1) ?

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 ?

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 ?

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.

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 ?

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 ?

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

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

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

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

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 ?

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

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

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

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 cha

What is the remainder if 2¹⁹² is divided by 6 ?

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

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 righ

A bill for ₹ 1,840 is paid in the denominations of ₹ 50, ₹ 20 and ₹ 10 notes. 50 notes in all are used. Consider the following statements : 1. 25 notes of ₹ 50 are used and the remaining are in the denominations of ₹ 20

Which number amongst 2⁴⁰, 3²¹, 4¹⁸ and 8¹² is the smallest?

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

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

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 13

What is the remainder when 91 × 92 × 93 × 94 × 95 × 96 × 97 × 98 × 99 is divided by 1261?

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?

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?

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 prim

If 15 × 14 × 13 × ⋯ × 3 × 2 × 1 = 3ᵐ × n where m and n are positive integers, then what is the maximum value of m?

If 3²⁰¹⁹ is divided by 10, then what is the remainder ?

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

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

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.

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 ?

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 ?

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 ?

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

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

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

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

In the sum ⊗ + 1⊗ + 5⊗ + ⊗⊗ + ⊗1 = 1⊗⊗ for which digit does the symbol ⊗ stand?

If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length can you measure?

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?

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

What is the largest number among the following?

What is the greatest length x such that 3½ m and 8¾ m are integral multiples of x?

The recurring decimal representation 1.272727... is equivalent to

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

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

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

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

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?

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

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

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 identif

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

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 ?

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

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

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

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

Consider the following sum : • + 1• + 2• + •3 + •1 = 21• In the above sum, • stands for

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

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 n

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

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

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?

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