### Question. 1

How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 4^{2017}?

How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 4^{2017}?

How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?

Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?

How many 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3?

If x and y are positive real numbers satisfying x + y = 102, then the minimum possible value of 2601(1 + 1x">1/x)(1 + 1y">1/y) is

Let teh m-th and n-thterms of a Geometric progression be 34">3/4 and 12, respectively, when m < n. If the common ratio of the progression is an integer r, then the smallest possible value of r + n - m is

A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of 5 children. How many toffees were there in his stock initially?

If y is a negative number such that 2^{y2log35} = 5^{log23}, then y equals

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

Among 100 students, x_{1} have birthdays in January, x_{2} have birthdays in February, and so on. If x_{0} = max(x_{1}, x_{2}, ..., x_{12}), then the smallest possible value of x_{0} is

The mean of all 4 digit even natural numbers of the form 'aabb', where a>0, is

If log_{4} 5 = (log_{4} y) (log_{6} √5), then y equals

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

If (2n+1) + (2n+3) + (2n+5) + ... + (2n+47) = 5280 , then what is the value of 1+2+3+ ... +n ?

The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 is

In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is

How many pairs (m,n) of positive integers satisfy the equation the equation m^{2} + 105 = n^{2}?

How many factors of 2^{4} x 3^{5} x 10^{4} are perfect squares which are greater than 1?

Let a_{1} , a_{2} be integers such that a_{1} - a_{2} + a_{3} - a_{4} + ........ +(-1)^{n-1} a_{n} = n , for n ≥ 1. Then a_{51} + a_{52} + ........ + a_{1023} equals

Let a, b, x, y be real numbers such that a^{2} + b^{2} = 25 , x^{2} + y^{2} = 169 and ax + by = 65. If k = ay - bx, then

The real root of the equation 2^{6x} + 2^{3x+2} - 21 = 0 is

Let x and y be positive real numbers such that log_{5}(x + y) + log_{5}(x - y) = 3, and log_{2}y - log_{2}x = 1 - log_{2}3. Then xy equals

If m and n are integers such that (2–√2)^{19} 3^{4} 4^{2} 9^{m} 8^{n} = 3^{n} 16^{m} (∜64) then m is

If a_{1} + a_{2} + a_{3} + ... + a_{n} = 3(2^{n+1} - 2), then a_{11} equals [TITA]

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is

If the population of a town is p in the beginning of any year then it becomes 3+2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

The arithmetic mean of x, y and z is 80, and that of x, y, z, u and v is 75, where u = (x+y)/2and v = (y+z)/2. If x ≥ z, then the minimum possible value of x is

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

If the sum of squares of two numbers is 97, then which one of the following cannot be their product

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ..... + 95 x 99 is

The smallest integer n such that n^{3} - 11n^{2} + 32n - 28 ＞ 0 is

The smallest integer n for which 4^{n} ＞ 17^{19} holds, is closest to

If p^{3} = q^{4} = r^{5} = s^{6}, then the value of log_{s} (pqr) is equal to

If N and x are positive integers such that N^{N} = 2^{160} and N^{2} + 2^{N} is an integral multiple of 2^{x}, then the largest possible x is

Let t_{1}, t_{2},… be real numbers such that t_{1}+ t_{2} +... + t_{n} = 2n^{2} + 9n + 13, for every positive integer n ≥ 2. If t_{k}=103, then k equals

The number of integers x such that 0.25 < 2^{x} < 200, and 2^{x} + 2 is perfectly divisible by either 3 or 4, is

Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is

How many numbers with two or more digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 so that in every such number, each digit is used at most once and the digits appear in the ascending order?

While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is

If a_{1} = 1/(2×5) , a_{2} = 1/(5×8), a_{3} = 1/(8×11),...., then a_{1} + a_{2} + a_{3} + ...... + a_{100} is

An infinite geometric progression a_{1}, a_{2}, a_{3},... has the property that a_{n} = 3(a_{n+1} + a_{n+2} +....) for every n ≥ 1. If the sum a_{1} + a_{2} + a_{3} +...... = 32, then a_{5} is

How many different pairs (a, b) of positive integers are there such that a ≤ b and 1/a + 1/b = 1/9 ?

Let a_{1}, a_{2}, a_{3}, a_{4}, a_{5} be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with 2a_{3}. If the sum of the numbers in the new sequence is 450, then a_{5} is

If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

The numbers 1, 2,..., 9 are arranged in a 3 X 3 square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value. If the top left and the top right entries of the grid are 6 and 2, respectively, then the bottom middle entry is

Let a_{1}, a_{2},.......a_{3n} be an arithmetic progression with a_{1} = 3 and a_{2} = 7. If a_{1} + a_{2} + ......+a_{3n} = 1830, then what is the smallest positive integer m such that m (a_{1} + a_{2} + ..... + a_{n}) > 1830?

If the square of the 7^{th} term of an arithmetic progression with positive common difference equals the product of the 3^{rd} and 17^{th} terms, then the ratio of the first term to the common difference is

A natural number n is such that 120 n<=240. If HCF of n and 240 is 1, how many values of n are possible?

If N = 888…up to 100 digits, what is the remainder when N is divided by 625?

A sequence of 4 digits, when considered as a number in base 10 is four times the number it represents in base 6. What is the sum of the digits of the sequence?

A four-digit number is divisible by the sum of its digits. Also, the sum of these four digits equals the product of the digits. What could be the product of the digits of such a number?

Out of 4 numbers a, b, c, and d, each pair of numbers has the same highest common factor. Find the highest common factor of all the four numbers if the least common multiple of a and b is 310 and that of c and d is 651.

How many elements are there in P?

An amount borrowed at simple interest gets tripled in 24 years. How many years does it take to get doubled, if the interest rate is same

From a vessel completely filled up with pure wine, 140 litres of content is removed and replaced with equal quantity of water. The process is repeated one more time. In a 98 litres sample of the resulting solution 80 litres is water. Find the capacity (in litres) of the vessel.

‘P’ is the product of ten consecutive two-digit natural numbers. If 2a is one of the factors of P, then the maximum value that ‘a’ can assume is

If 7^a = 26 and 343^b = 676 then what is the relation between a and b?

A set ‘P’ is formed from the set of first ‘N’ natural numbers by deleting all the perfect squares and all the perfect cubes. If the numbers are arranged in an ascending order then, what is the 476th number of the set ‘P’?

. Find the number of ways in which a batsman can score 100 runs by scoring runs in 2’s, 4’s and 6’s, such that he hits at least one double, one boundary and one six.

The ratio of two numbers whose is 600 is 7 : 8. What is the LCM of the given two numbers?

The number of APs with 5 distinct terms that can be formed from the first 50 natural numbers is

How many natural numbers divide exactly one out of 1080 and1800, but not both?

The number of factors of the square of a natural number is 105. The number of factors of the cube of the same number is ‘F’. Find the maximum possible value of ‘F’.

‘ab’ is a two-digit prime number such that one of its digits is 3. If the absolute difference between the digits of the number is not a factor of 2, then how many values can ‘ab’ assume?

If E = 3 + 8 + 15 + 24 + … + 195, then what is the sum of the prime factors of E?

The number 44 is written as a product of 5 distinct integers. If ‘n’ is the sum of these five integers then what is the sum of all the possible values of n?

500! + 505! + 510! + 515! is completely divisible by 5^n , where n is a natural number. How many distinct values of n are possible?

All the two-digit natural numbers whose unit digit is greater than their ten’s digit are selected. If all these numbers are written one after the other in a series, how many digits are there in the resulting number?

(x – 1)(x – 2)(x – 3) = 6^y .How many integer solutions exist for the given equation?

(X+3)/3 ,(X+8)/4 , (X+15)/5 , (X+24)/6,... , (X+80)/10 is a sequence where X is not equal to 1. What is the least value of X for which HCF (Numerator, Denominator) = 1 for each term of the given sequence?

The last digit of 3^3^4n , is

What is the number of non-negative integer solutions for the equation x2 – xy + y2 = x + y?

If ‘a’ is one of the roots of x5– 1 = 0 and a ¹1, then what is the value of a15 + a16 + a17 +.......a50?

If x and y are positive integers, then the last digit of which of the following is same as the last digit of the sum of x and y?

A positive integer is equal to the square of the number of factors it has. How many such integers are there?

P, Q and R are three consecutive odd numbers in ascending order. If the value of three times P is three less than two times R, find the value of R.

If 7103 is divided by 25, then the remainder is

If S = 12 – 22 + 32 – 42 + ..... – 20002 + 20012 , then what is the value of S?

N! is completely divisible by 1352.What is sum of the digits of the smallest such number N?

How many numbers are there between 0 and 1000 which on division by 2, 4, 6, 8 leave remainders 1, 3, 5, 7 respectively?

What are the last two digits of 7^{2008} ?

How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

The integers 1, 2, ...., 40 are written on a blackboard. The following operation is then repeated 39 times; In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b – 1 is written. What will be the number left on the board at the end?

How many pairs of positive integers m, n satisfy 1/m + 4/n = 1/12 where n is an odd integer less than 60?

Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

If x = –0.5, then which of the following has the smallest value?

For a positive integer n, let p_{n} denote the product of the digits of n, and s_{n} denote the sum of the digits of n. The number of integers between 10 and 1000 for which p_{n} + s_{n} = n is

The rightmost non-zero digit of the number 30^{2720} is

The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B–A is perfectly divisible by 7, then which of the following is necessarily true?

If x = (16³ + 17³ + 18³ + 19³ ), then x divided by 70 leaves a remainder of

The remainder, when (15^{23} + 23^{23}) is divided by 19, is

If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is

Let a, b, c, d and e be integers such that a = 6b = 12c, and 2b = 9d = 12e. Then which of the following pairs contains a number that is not an integer?

Let n( > 1) be a composite integer such that √n is not an integer. Consider the following statements

I : n has a perfect integer-valued divisor which is greater than 1 and less than √n .

II : n has a perfect integer-valued divisor which is greater than √n but less than n

Then,

Let x and y be positive integers such that x is prime and y is composite. Then

What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7?

The seven basic symbols in a certain numeral system and their respective values are as follow :

I = 1 , V = 5 , X = 10, L = 50, C = 100, D = 500, and M = 1000

In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example , XXVII = 10 + 10 + 5 + 1 + 1= 27. An exception to the left to right reading occurs when a symbol is followed immediately by a symbol of greater value; then, the smaller value is subtracted from the larger.

For example, XLVI = (50 – 10) + 5 + 1 = 46.

The value of the numeral MCMXCIX is

The seven basic symbols in a certain numeral system and their respective values are as follow :

I = 1 , V = 5 , X = 10, L = 50, C = 100, D = 500, and M = 1000

In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example , XXVII = 10 + 10 + 5 + 1 + 1= 27. An exception to the left to right reading occurs when a symbol is followed immediately by a symbol of greater value; then, the smaller value is subtracted from the larger.

For example, XLVI = (50 – 10) + 5 + 1 = 46.

Which of the following can represent the numeral for 1995?

I. MCMLXXV

II. MCMXCV

III. MVD

IV. MVM

The seven basic symbols in a certain numeral system and their respective values are as follow :

I = 1 , V = 5 , X = 10, L = 50, C = 100, D = 500, and M = 1000

In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example , XXVII = 10 + 10 + 5 + 1 + 1= 27. An exception to the left to right reading occurs when a symbol is followed immediately by a symbol of greater value; then, the smaller value is subtracted from the larger. For example, XLVI = (50 – 10) + 5 + 1 = 46.

The value of the numeral MDCCLXXXVII is

What is the remainder when 4^{96} is divided by 6?

The number of positive integers n in the range 12 ≤ n ≤ 40 such that the product (n-1)(n-2).....3.2.1 is not divisible by n is

How many even integers n, where 100 ≤ n ≤ 200, are divisible neither by seven nor by nine?

A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals

7^{6n} – 6^{6n}, where n is an integer > 0, is divisible by

If u, v, w and m are natural numbers such that u^{m} + v^{m} = w^{m}, then one of the following is true

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

At a book store, “MODERN BOOK STORE” is flashed using neon lights. The words are individually flashed at intervals of 2(1/2), 4(1/4), 5(1/8) seconds respectively, and each word is put off after a second. The least time after which the full name of the bookstore can be read again is

When 2^{256} is divided by 17 the remainder would be

Number S is obtained by squaring the sum of digits of a two digit number D. If difference between S and D is 27, then the two digit number D is

Let x, y and z be distinct integers. x and y are odd positive, and z is even positive. Which one of the following statements can not be true?

Every ten years the Indian government counts all the people living in the country. Suppose that the director of the census has reported the following data on two neighbouring villages Chota Hazri and Mota Hazri

Chota Hazri has 4,522 fewer males than Mota hazri.

Mota Hazri has 4,020 more females than males.

Chota Hazri has twice as many females as males.

Chota Hazri has 2,910 fewer females than Mota Hazri.

What is the total number of males in Chota Hazri?

In a number system the product of 44 and 11 is 3414. The number 3111 of this system, when converted to the decimal number system, becomes

Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

In a 4 - digit number, the sum of the first two digits is equal to that of the last two digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other two digits. What is the third digit of the number?

Out of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The number of boxes containing the same number of oranges is at least

55³ +17³ - 72³ is divisible by

If x, y and z are odd integers then which of the following is necessarily false?

There are two integers 34041 and 32506, when divided by a three-digit integer n, leave the same remainder. What is the value of n?

N = 1421 × 1423 × 1425 what is the remainder when N is divided by 12?

P is the product of all the prime numbers between 1 to 100. Then the number of zeroes at the end of P are

Let D be a recurring decimal of the form D = 0.a1 a2 a1 a 2 a 1 a 2 .............where a1 and a2 lie between 0 and 9. Further at most one of them is zero. Which of the following numbers necessarily produces an integer when multiplied by D?

Convert 1982 in base 10 to base 12

For the given pair (x, y) of positive integers, such that 4x – 17y = 1 and x ≤ 1000, how many integer values of y satisfy the given conditions

A, B, C are three distinct digits. AB is a two digit number and CCB is a three digit number such that (AB)² = CCB where CCB > 320. What is the possible value of the digit B?

When 7^{84} is divided by 342, what is the remainder?

n² = 12345678987654321, then, n = ?

If n = 1 + x, where ‘x’ is the product of four consecutive positive integers, then which of the following statements is/are true?

I. ‘n’ is odd

II. ‘n’ is prime

III. ‘n’ is perfect square

A hundred digit number is formed by writing first 54 natural numbers in front of each other as 12345678910111213.................5354. Find the remainder when this number is divided by 8

What is the digit in the unit place of 2^{51} ?

Number of students who have opted the subjects A, B, C are 60, 84, 108 respectively. The examination is to be conducted for these students such that only the students of the same subject are allowed in one room. Also the number of students in each room must be same. What is the minimum number of rooms that should be arranged to meet all these conditions?

A is the set of positive integers such that when divided by 2,3,4,5 and 6 leaves the remainders 1,2,3,4 and 5 respectively. How many integer(s) between 0 and 100 belongs to set A?

A certain number when divided by 899 leaves the remainder 63. Find the remainder when the same number is divided by 29.

Three wheels can complete respectively 60,36,24 revolutions per minute. There is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch the ground again?

(BE)² = MPB, where B, E, M and P are distinct integers, then M = ?

Five digit numbers are formed using only 0,1,2,3,4 exactly once. What is the difference between the maximum and minimum number that can be formed?

n³ is odd. Which of the following statements is/are true?

I. n is odd

II. n² is odd

III. n² is even

P and Q are two integers such that (PQ) = 64. Which of the following cannot be the value of P + Q?

P, Q and R are three consecutive odd numbers in ascending order. If the value of three times P is three less than two times R, find the value of R.

Which of the following is true?

If n is an integer, how many values of n will give an integral value of (16n² + 7n + 6) / n?

A student, instead of finding the value of 7/8th of a number, found the value of 7/18th of the number. If his answer differed from the actual one by 770, find the number.

If m and n are integers divisible by 5, which of the following is not necessarily true?

If a number 774958A96B is to be divisible by 8 and 9, the values of A and B, respectively, will be

If n is any odd number greater than 1, then n(n² – 1) is

A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. Once, by mistake, both the numbers were entered with their digits interchanged. The total sales value remained the same, i.e. Rs 1148, but the inventory reduced by 54.

What is the actual price per piece ?

A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. Once, by mistake, both the numbers were entered with their digits interchanged. The total sales value remained the same, i.e. Rs 1148, but the inventory reduced by 54.

What is the actual quantity sold ?

5^{6} – 1 is divisible by

Three consecutive positive even numbers are such that thrice the first number exceeds double the third number by 2 then the third number is

Cost of 72 hens is Rs .....96.7..... Then, what will be the cost of hen, where two digits in place of “ ....... ” are not visible or are written in illegible hand-writing?

The remainder obtained when a prime number greater than 6 is divided by 6 is

Two positive integers differ by 4 and sum of their reciprocals is 10/21. Then one of the number is

Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together. What length of time will elapse before they chime together again?

For the product n(n+1)(2n+1), n ∈ N, which one of the following is not necessarily true?

What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?

Which is the least number that must be subtracted from 1856, so that the remainder when divided by 7, 12 and 16 will leave the same remainder 4