### 12

 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. 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 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.