### 13

Divisibility by 13

Method 1:

The number is divisible by 13 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 13.
Remember the sequence 1, 10, 9, 12, 3, 4, 1, 10, 9, 12, 3, 4, 1, 10, 9, 12,....
Here 1, 10, 9, 12, 3, 4, repeat in 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 8×9 3×12 5×3 6×4 2×1 4×10
-- 6 70 72 26 15 24 2 40
2.Add the numbers 6+70+72+26+15+24+2+40=265
3.Repeat step 0
5 6 2 0 0 0 0 0
4.Multiply with
the sequence
5×1 6×10 2×9 0 0 0 0 0
-- 5 60 18 0 0 0 0 0
5+60+18 =83

7.Divide the result by 13
find the remainder
mod(83/13)=5
If the remainder is zero (0) then the number is divisible by 13 else the result is remainder.

The remainder is 5. 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 13 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 13.
Remember the sequence 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4,  -1, ....
Here 1, -3, -4, -1, 3, 4 repeat in 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×-3 8×-4 3×-1 5×3 6×4 2×1 4×-3
-- 6 -21 -32 -3 15 24 2 -12
2.Add the numbers 6-21-32-3+15+24+2-12=-21
The number is not divisible by 13.
If the number is a multiple of 13 then the number is divisible by 13 else not.

Remainder

1. In method 1 we can surely say that the result is the remainder of the original number.
2. In method 2 we are not sure. If the result is <13 and >0 then the number is the remainder else 13+ negative number is the remainder. -12≤negative number≤-1.