# Divisibilty

Type 1: BASIC DIVISIBILITY

1. 7386038 is divisible by: [CDS 2001, 02]

(a) 3

(b) 4

(c) 9

(d) 11

Explanation:

The given number is 7386038.

The sum of the digits at even places is = 3+ 6+ 3=12.

The sum of the digits at odd places is = 7+ 8+ 0+ 8 = 23.

The difference is given by 23-12 = 1

2. Which one of the following is divisible by 9? [CDS 1999]

(a) 23, 50, 821

(b) 28, 70, 052

(c) 42, 13, 533

(d) 64, 000, 80

Explanation:

(a) 2350821 = 2+3+5+0+8+2+1 = 21, not divisible by 9

(b) 2870052 = 2+8+7+5+2 = 24, not divisible by 9.

(c) 4213533 = 4+2+1+3+5+3+3 = 19, not divisible by 9.

(d) 6400080 = 6+4+8 = 18, divisible by 9.

So, the answer is (d) 64000

3. Which one of the following numbers is divisible by 11? [CDS 2001]

(a) 4823718

(b) 8423718

(c) 8432718

(d) 4832718

Explanation:

4832718 = (4+3+7+8) – (8+2+1) = 11

So, the given number is divisible by 11

4. Which one of the following numbers is divisible by 99? [CDS 2002, 01]

(a) 3572404

(b) 135792

(c) 913464

(d) 114345

Explanation:

The sum of all the digits of 114345 is 18.

The sum of all digits at even places and odd places are 9 and 9 respectively.

The difference between the sums is 0.

So, the number is divisible by both 9 and 11 and hence by 99.

Type 2: DIVISIBILITY WITH ADDITION AND SUBTRACTION

5. The sum of three consecutive natural numbers each divisible by 3 is 72. What is the largest among them? [SSC Grad. 1999]

(a) 21

(b) 24

(c) 27

(d) 30

Explanation:

Let us take three consecutive numbers which are divisible by 3 are, 3x, 3x+3 and 3x+6.

Consider x as 7,

The largest of them= 27.

6. In a six digit number, the sum of the digits in the even places is 9 and the sum of the digits in the odd places is 20. All such numbers are divisible by:

(a) 7

(b) 9

(c) 6

(d) 11

Explanation:

Given that the sum of the digits at even places is 9 and that of at odd places is 20.

The difference between the two sums is 20-9=11.

So, all such numbers are divisible by 11

7. If the number 7×86038 is exactly divisible by 11, then the smallest whole number in place of x?

(a) 2

(b) 3

(c) 1

(d) 4

Explanation:

The given number =7×86038

Sum of the odd places =8+0+8+7=23

Sum of the even places = 3+6+x

(Sum of the odd places)- (Sum of even places) = Number (exactly divisible by 11)

23-(9+x) = divisible by 11

14 – x = divisible by 11.

X must be 3, to make given number divisible by 11.

8. Which one of the following numbers will completely divide by ?

(a) 7

(b) 11

(c) 9

(d) 13

Explanation:

9. The total number of two digit positive integer < than 100, which are not divisible by 2, 3 and 5 is

(a) 23

(b) 24

(c) 25

(d) 26

Explanation:

Numbers less than 100 divisible by 2 are 10, 12, 14, 16,… 98

Total numbers divisible by 2 are 45

Remaining numbers are 11, 13, 15, 17, 19, … 99 = 45 integers

Numbers divisible by 3 in the above numbers

15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93 and 99

Total numbers divisible by 3 are = 15

Remaining integers = 45 – 15 = 30

Numbers divisible by 5 in the remaining numbers = 25, 35, 55, 65, 85, 95

Total numbers = 6

Remaining 30–6 = 24 numbers are not divisible by 2, 3 and 5

10.  When n is an integer > 0 is divisible by

(a)  120

(b) 127

(c) 130

(d) 137