Divisibility by Six
 Method 1:
 If all the digits except the digit at the unit place are added.
 The result is multiplied by 4.
 The result of step 2 is added to the digit at the unit place.
 If result of step 3 is large then the step 1, 2 and 3 is repeated in order.
 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. 10 ^{x} 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
 Is 340 divisible by 6.
 (3+4)×4 + 0 = 28
 28
 Remainder when 28 is divided by 6 = 4
 The number is not divisible and the remainder is 4.
 Is 3483 divisible by 6.
 (3+4+8)×4 + 3 = 63
 63
 Repeat the process , 6×4+3 = 27
 Remainder when 27 is divided by 6 = 3
 The number is not divisible and the remainder is 3.
