# Divisibility by Six

• Method 1:
1. If all the digits except the digit at the unit place are added.
2. The result is multiplied by 4.
3. The result of step 2 is added to the digit at the unit place.
4. If result of step 3 is large then the step 1, 2 and 3 is repeated in order.
5. If the result of 4 is divided by 6 and the remainder is zero then the number is divisible by 6 else not.
• Method 2:⇒ If the number is divisible by 2 and 3 both then the number is divisible by 6.

## Proof, How: Division by 6

Let the number be abcd.

abcd can be written as
a×1000 + b×100 + c×10 + d

which can also be written as
a×(6×166+4) + b×(6×16+4) + c×(6+4) + d

Every power of 10 i.e. 10x when divided by 6 gives 4 as remainder.
Reason: 10 = 6 + 4

100 = 10(6+4)
= 10×6 + 10×4 , here remainder is remainder of 40.

1000 = 100(6+4)
= 100×6 + 100×4
= 100×6 + 4×10×10 = 100×6 + 4×10×(6+4)
= 100×6 + 4×6×10 + 4×4

above has remainder 4 when divided by 6.

The process continues and we get 4 as remainder for every power of 10.

= a×(6×166+4) + b×(6×16+4) + c×(6+4) + d Every product is divisible except a×4 + b×4 + c×4 + d

So the number is divisible if the sum k is divisible by 6.

The number is divisible by 6 if [(sum of all digits except at unit)×4 + digit at unit] is divisible by 6.

## Remainder on division by 6

• The remainder is the remainder obtained in method 1.

## Examples of division by 6

1. Is 340 divisible by 6.
1. (3+4)×4 + 0 = 28
2. 28
3. Remainder when 28 is divided by 6 = 4
4. The number is not divisible and the remainder is 4.

2. Is 3483 divisible by 6.
1. (3+4+8)×4 + 3 = 63
2. 63
3. Repeat the process , 6×4+3 = 27
4. Remainder when 27 is divided by 6 = 3
5. The number is not divisible and the remainder is 3.