### 14

Divisibility by 14

Method 1:

The number is divisible by 14 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 14.
Remember the sequence 1, 10, 2, 6, 4, 12, 8, 10, 2, 6, 4, 12, 8, 10, 2,....
Here 10, 2, 6, 4, 12, 8 repeat in the sequence 1 is added only at the beginning of the sequence.
0.Reverse the order of
the digits
6 7 8 3 5 6 2 4
1.Multiply with
the sequence
6×1 7×10 8×2 3×6 5×4 6×12 2×8 4×10
-- 6 70 16 18 20 72 16 40
3.Repeat step 0 with
8 5 2 0 0 0 0 0
4.Multiply with
the sequence
8×1 5×10 2×2 0 0 0 0 0
-- 8 50 4 0 0 0 0 0
8+50+4 =62

7.Divide the result by 14
find the remainder
mod(62/14)=6
If the remainder is zero (0) then the number is divisible by 14 else the result is remainder.

The remainder is 6. Repeat the step 0,1 and 2 till you get a number whose remainder you can find easily.

Method 2:

The number is divisible by 14 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 14.
Remember the sequence 1, -4, 2, 6, 4, -2, -6, -4, 2, 6, 4, -2, -6, -4, ....
Here -4, 2, 6, 4, -2, -6 repeat in the sequence, 1 is added only at the beginning of the sequence.
0.Reverse the order of
the digits
6 7 8 3 5 6 2 4
1.Multiply with
the sequence
6×1 7×-4 8×2 3×6 5×4 6×-2 2×-6 4×-4
-- 6 -28 16 18 20 -12 -12 -16