Divisibility by 12
- Method 1:
- If all the digits except the digit at the unit place and tens place are added.
- The result is multiplied by 4.
- The result of step 2 is added to 10 times the digit at tens place.
- The result of step 3 is added to the digit at the unit place.
- If result of step 4 is large then the step 1, 2, 3 and 4 is repeated in order.
- If the result of step 5 is divided by 12 and the remainder is zero then the number is divisible by 12 else not.
- Method 2:⇒ If the number is divisible by 4 and 3 both then the number is divisible by 12.
Proof:
Let the number be abcde.
abcde can be written as
a×10000 + b×1000 + c×100 + d×10 + e
which can also be written as
a×(12×833+4) + b×(12×83+4) + c×(12×8+4) + d×10 + e
Every power of 10 i.e. 10^{x} when divided by 12 gives 4 as remainder,x>1 and is a positive integer.
Every product is divisible except a×4 + b×4 + c×4 + d×10 + e
So the number is divisible if the sum is divisible by 12.
The number is divisible by 12 if [(sum of all digits except the last two digits from right)×4 + 10×(digit at the tens place)+ digit at unit place ] is divisible by 12.
Remainder
- The remainder is the remainder obtained in method 1.
Examples
- Is 340 divisible by 12.
- 3×4 + 4×10 + 0 = 52
- 52
- Remainder when 52 is divided by 12 = 4
- The number is not divisible and the remainder is 4.
- Is 3483 divisible by 12.
- (3+4)×4 + 8×10 + 3 = 111
- 111
- Repeat the process , 1×4+10+1 = 15
- Remainder when 15 is divided by 12 = 3
- The number is not divisible and the remainder is 3.