Number Systems

While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred's place is greater than the digit at ten's place, and the digit at ten's place is greater than the digit at unit's place?

1. 61
2. 64
3. 85
4. 91

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p² - 1) is

1. always divisible by 6, and may or may not be divisible by 12
2. always divisible by 24
3. never divisible by 6
4. always divisible by 12, and may or may not be divisible by 24

If n is a positive integer, then (√3+1)²ⁿ - (√3−1)²ⁿ is

1. An even positive integer
2. A rational number other than positive integers
3. An odd positive integer
4. An irrational number

The last digit of the number 3²⁰¹⁵ is

1. 1
2. 3
3. 5
4. 7

When we multiply a certain two-digit number by the sum of its digits, 405 is achieved. If you multiply the number written in reverse order of the same digits, we get 486. Find the number?

1. 81
2. 45
3. 36
4. 54

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?

1. 1
2. 2
3. 3
4. 4

The number of integers n satisfying -–n+2 ≥ 0 and 2n ≥ 4 is

1. 0
2. 1
3. 2
4. 3

What is the right most non-zero digit of the number 30²⁷²⁰?

1. 1
2. 3
3. 7
4. 9

The product of all integers from 1 to 100 will have the following numbers of zeros at the end?

1. 19
2. 20
3. 22
4. 24

The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?

1. 21
2. 25
3. 41
4. 67

The number of positive integer valued pairs (x, y) satisfying 4x - 17y = 1 and x ≤ 1000 is

1. 55
2. 57
3. 58
4. 59

What are the last two digits of 7²⁰⁰⁸?

1. 01
2. 21
3. 41
4. 61

What is the digit in the unit’s place of 2⁵¹?

1. 1
2. 2
3. 4
4. 8

When you reverse the digits of the number 13, the number increases by 18. How many other two-digit numbers increase by 18 when their digits are reversed?

1. 5
2. 6
3. 7
4. 8

How many times will the digit 5 come in counting from 1 to 99 excluding those which are divisible by 3?

1. 16
2. 15
3. 14
4. 13

What is the number of all possible positive integer values of n for which n² + 96 is a perfect square?

1. 2
2. 4
3. 5
4. Infinite

If N = (11^{p + 7})(7^{q – 2})(5^{r + 1})(3^s) is a perfect cube, where p, q, r and s are positive integers, then the smallest value of p + q + r + s is:

1. 5
2. 6
3. 7
4. 8
5. 9

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

1. 1777
2. 1785
3. 1875
4. 1877

Let f(x) = x² and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

1. 16
2. 18
3. 36
4. 40