*Tricks for Divisibility*

* *

*Some basics tips and tricks that you can use for divisibility*

All whole numbers are divisible by 1.

A number is divisible by 2 if it’s even.

A non-zero number is divisible by 5 if it ends in 0 or 5.

In order to check the divisibility of a number by a composite number, divide the composite divisor into prime factors, which are co-prime and then check for its divisibility with each. For example, to check the divisibility of a number with 12, break down 12 into 3 and 4.

**Tricks for Divisibility**

**a ^{n} – b^{n} is always divisible by a-b**

**8**–

^{5}**5**is divisible by 8-5= 3

^{5}Remember it by:

**a ^{3}**–

**b**is divisible by a-b

^{3}**a**–

^{2}**b**is also divisible by a-b

^{2}**a**

^{n}– b^{n}is divisible by a+b when n is even**7**is divisible by 7-5= 2

^{10}– 5^{10}Remember it by:

a

^{3}– b

^{3}is not divisible by a+b

a

^{2}– b

^{2}is also divisible by a+b

a

^{4}– b

^{4}is also divisible by a+b

**a ^{n} + b^{n }is divisible by a+b when n is odd**

**7**is divisible by 7+5= 12

^{11}+ 5^{11}Remember it by:

a

^{3}+ b

^{3}is divisible by a+b

a

^{2}+ b

^{2}is NOT divisible by a+b

a

^{4}+ b

^{4}is NOT divisible by a+b

**a ^{n} + b^{n}+c^{n}is divisible by a+b+c when n is odd.**

7

^{3}+ 5

^{3}+ 2

^{3}= 343+125+8=476 divisible by 7+5+2=14

*Questions you can solve by using above formula:*

**Example 1: 32 ^{23} + 17^{23}is definitely divisible by….**

a. 49

b. 15

c. 49 & 15

d. none of these.

ANSWER A

As

**a**

^{n}+ b^{n }is divisible by a + b when n is oddSo 32

^{23}+ 17

^{23}is divisible by 32 + 17 = 49.

**Example 2: 32 ^{23} – 17^{23} is definitely divisible by….**

a. 49

b. 15

c. 49 & 15

d. none of these.

ANSWER B

As

**a**

^{n}– b^{n}is always divisible by a-b.So 32

^{23}– 17

^{23}is divisible by 32 – 17 = 15.

**Example 3: 32 ^{232} – 17^{232} is definitely divisible by….**

a. 49

b. 15

c. 49 & 15

d. none of these.

ANSWER C

As **a ^{n} – b^{n} is always divisible by a-b** and

**a**

^{n}– b^{n}is divisible by a + b when n is evenSo 32

^{232}– 17

^{232}is divisible by both 32 – 17 = 15 and 32 + 17 = 49.

**Example 4: 3 ^{5} + 5^{5}+ 7^{5}is definitely divisible by….**

a. 8

b. 7

c. 15

d. all of these.

ANSWER C

As

**a**

^{n}+ b^{n }+ c^{n }is divisible by a + b + c when n is odd.So 3

^{5}+ 5

^{5}+ 7

^{5}is divisible by 3 + 5 + 7 = 15