1. 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?

a) 80

b) 76

c) 41

d) 53

Explanation: According to the question the required no. is

3[4 (7x + 4) + 1] + 2 = 84x + 53

So the remainder is 53, when the same number is divided by 84.

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

a) \[m \geq min\left(u, v,w\right)\]

b) \[m \geq max\left(u, v,w\right)\]

c) \[m < min\left(u, v,w\right)\]

d) None of these

Explanation:

3. \[7^{6n}-6^{6n}\] , where n is an integer > 0, is divisible by

a) 13

b) 127

c) 559

d) All of these

Explanation:

4. 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

a) 31

b) 63

c) 75

d) 91

Explanation:

5. How many even integers n, where \[100 \leq n\leq 200\] , are divisible neither by seven nor by nine?

a) 40

b) 37

c) 39

d) 38

Explanation:

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

a) 5

b) 7

c) 13

d) 14

Explanation: Consider the prime numbers between 12 and 40, which are 13, 17, 19, 23, 29, 31 and 37.

Given product is not divisible by these 7 prime numbers

7.What is the remainder when \[4^{96}\] is divided by 6?

a) 0

b) 2

c) 3

d) 4

Explanation:

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

a) 666

b) 676

c) 683

d) 777

Explanation:

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

a) y – x cannot be an even integer

b) xy cannot be an even integer

c) (x + y)/x cannot be an even integer

d) None of the above statements is true

Explanation:

10. Let n( > 1) be a composite integer such that \[\sqrt{n}\] is not an integer. Consider the following statements

I : n has a perfect integer-valued divisor which is greater than 1 and less than \[\sqrt{n}\] .

II : n has a perfect integer-valued divisor which is greater than \[\sqrt{n}\] but less than n

Then,

a) Both I and II are false

b) I is true but II is false

c) I is false but II is true

d) Both I and II are true

Explanation: