CAT 2025 Lesson : Class Discussion: Numbers - Divisibility: 1 to 8

Class Discussion - Divisibility

Question 1

Consider all 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?
[CAT 2007]

		3
		2
		4
		0
		1

Question 2

What is the remainder when 62626262...

_{200 \ \mathrm{digits}}

is divided by 36?

		0
		6
		16
		26

Question 3

What is the remainder when

113^{112}

is divided by 25?

		1
		6
		16
		21

Question 4

Anand multiplies 4 consecutive odd natural numbers, then divides the product by 10 and notes the remainder. Aravind multiplies 4 consecutive even natural numbers, divides the product by 10 and notes the remainder. Which of the following numbers is definitely not the remainder calculated by either of them?

		3
		4
		5
		9

Question 5

a

and

b

are prime numbers and,

c

and

d

are natural numbers, such that

ac

when divided by

b

yields a remainder of 1 and

bd

when divided by

a

yields a remainder of 1. The remainder when

ac + bd

is divided by

ab

is always:

		0
		1
		$ab - 1$
		$\dfrac{ab -1}{2}$

Question 6

A set comprises of all possible products of five consecutive natural numbers, with each being less than 100. What is the HCF of all the elements in the set?

Answer: 120

Question 7

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
[CAT 2003 L]

		5
		7
		13
		14

Question 8

What is the greatest power of 189 that would perfectly divide 200!?

Answer: 32

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