12

Divisibility by 12

  • Method 1:
    1. If all the digits except the digit at the unit place and tens place are added.
    2. The result is multiplied by 4.
    3. The result of step 2 is added to 10 times the digit at tens place.
    4. The result of step 3 is added to the digit at the unit place.
    5. If result of step 4 is large then the step 1, 2, 3 and 4 is repeated in order.
    6. 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. 10x 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

  1. Is 340 divisible by 12.
    1. 3×4 + 4×10 + 0 = 52
    2. 52
    3. Remainder when 52 is divided by 12 = 4
    4. The number is not divisible and the remainder is 4.

  2. Is 3483 divisible by 12.
    1. (3+4)×4 + 8×10 + 3 = 111
    2. 111
    3. Repeat the process , 1×4+10+1 = 15
    4. Remainder when 15 is divided by 12 = 3
    5. The number is not divisible and the remainder is 3.


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