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
aⁿ – bⁿ is always divisible by a-b
8⁵–5⁵ is divisible by 8-5= 3

Remember it by:
a³– b³ is divisible by a-b
a²– b² is also divisible by a-b
aⁿ – bⁿ is divisible by a+b when n is even
7¹⁰ – 5¹⁰ is divisible by 7-5= 2
Remember it by:
a³ – b³ is not divisible by a+b
a² – b² is also divisible by a+b
a⁴ – b⁴ is also divisible by a+b

aⁿ + bⁿ is divisible by a+b when n is odd
7¹¹ + 5¹¹ is divisible by 7+5= 12
Remember it by:
a³ + b³ is divisible by a+b
a² + b²is NOT divisible by a+b
a⁴ + b⁴ is NOT divisible by a+b

aⁿ + bⁿ+cⁿis divisible by a+b+c when n is odd.
7³ + 5³ + 2³= 343+125+8=476 divisible by 7+5+2=14

Questions you can solve by using above formula:
Example 1: 32²³ + 17²³is definitely divisible by….

a. 49
b. 15
c. 49 & 15
d. none of these.
ANSWER  A
As aⁿ + bⁿ is divisible by a + b when n is odd
So 32²³ + 17²³ is divisible by 32 + 17 = 49.

Example 2: 32²³ – 17²³ is definitely divisible by….
a. 49
b. 15
c. 49 & 15
d. none of these.
ANSWER B
As aⁿ – bⁿ is always divisible by a-b.
So 32²³ – 17²³ is divisible by 32 – 17 = 15.

Example 3: 32²³² – 17²³² is definitely divisible by….
a. 49
b. 15
c. 49 & 15
d. none of these.
ANSWER C

As aⁿ – bⁿ is always divisible by a-b and aⁿ – bⁿ is divisible by a + b when n is even
So 32²³² – 17²³² is divisible by both 32 – 17 = 15 and 32 + 17 = 49.

Example 4: 3⁵ + 5⁵+ 7⁵is definitely divisible by….
a. 8
b. 7
c. 15
d. all of these.
ANSWER C
As aⁿ + bⁿ + cⁿ is divisible by a + b + c when n is odd.
So 3⁵ + 5⁵ + 7⁵ is divisible by 3 + 5 + 7 = 15