For any natural number $n$ , let $a_n$ be the largest integer not exceeding $\sqrt{n}$ . Then the value of $a_1+a_2+\cdots+a_{50}$ is

When $10^{100}$ is divided by 7 , the remainder is

If $m$ and $n$ are natural numbers such that $n>1$, and $m^n=2^{25} \times 3^{40}$, then $m-n$ equals

When $3^{333}$ is divided by 11 , the remainder is

If $10^{68}$ is divided by 13 , the remainder is

Let $n$ be the least positive integer such that 168 is a factor of $1134^n$ . If $m$ is the least positive integer such that $1134^n$ is a factor of $168^m$ , then $m+n$ equals

For any natural numbers $m$, $n$, and $k$, such that $k$ divides both $m + 2n$ and $3m + 4n$, $k$ must be a common divisor of

Let $a$, $b$, $m$, and $n$ be natural numbers such that $a > 1$, $b > 1$, and $a^m \cdot b^n = 144^{145}$, ten the largest possible value of $n - m$ is

The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

Let $A$ be the largest positive integer that divides all the numbers of the form $3^k+4^k+5^k$ , and $B$ be the largest positive integer that divides all the numbers of the form $4^k+3\left(4^k\right)+4^{k+2}$ , where $k$ is any positive integer. Then $(A+B)$ equals

For natural numbers $x$, $y$, and $z$, if $xy + yz = 19$ and $yz + xz = 51$, then what is the minimum possible value of $xyz$?

For any real number $x$ , let $[x]$ be the largest integer less than or equal to $x$ . If $\sum_{n=1}^N\left[\frac{1}{5}+\frac{n}{25}\right]=25$ then $N$ is

The number of integers greater than 2000 that can be formed with the digits 0, 1, 2, 3, 4, 5, using each digit at most once, is

For some natural number $n$ , assume that $(15,000)$ ! is divisible by $(n !) !$ . The largest possible value of $n$ is

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?

For a 4-digit number, the sum of its digits in the thousands, hundreds and tens places is 14, the sum of its digits in the hundreds, tens and units places is 15, and the tens place digit is 4 more than the units place digit. Then the highest possible 4-digit number satisfying the above conditions is

A box has 450 balls, each either white or black, there being as many metallic white balls as metallic black balls. If 40% of the white balls and 50% of the black balls are metallic, then the number of non-metallic balls in the box is

In a tournament, a team has played 40 matches so far and won 30% of them. If they win 60% of the remaining matches, their overall win percentage will be 50%. Suppose they win 90% of the remaining matches, then the total number of matches won by the team in the tournament will be

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

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

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

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

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 integers in the set {100, 101, 102, ..., 999} have at least one digit repeated?

How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 42017 ?

How many factors of $2^4 \times 3^5 \times 10^4$ are perfect squares which are greater than 1? [TITA]

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 [TITA]

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

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 [TITA]

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: [TITA]

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

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

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

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 a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)² + (a - c)² + (a - d)² is (TITA)

An elevator has a weight limit of 630 kg. It is carrying a group of people of whom the heaviest weighs 57 kg and the lightest weighs 53 kg. What is the maximum possible number of people in the group? [TITA]

The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive integers such that x ≤ 40, y ≤ 12, and z ≤ 12 is

How many different pairs $(a, b)$ of positive integers are there such that $a \leq b$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{9}$?

Let a₁, a₂, a₃, a₄, a₅ be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with 2a₃. If the sum of the numbers in the new sequence is 450, then a₅ is [TITA]

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