### 19

Divisibility by 19

Method 1:

The number is divisible by 19 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 19.
Remember the sequence 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2, 1, 10, 5, 12, 6,....
Here 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2 repeat in the sequence.
0.Reverse the order of
digits
6 7 8 3 5 6 2 4
1.Multiply with
the sequence
6×1 7×10 8×5 3×12 5×6 6×3 2×11 4×15
-- 6 70 40 36 30 18 22 60
3.Repeat step 0 with
2 8 2 0 0 0 0 0
4.Multiply with
the sequence
2×1 8×10 2×5 0 0 0 0 0
-- 2 80 10 0 0 0 0 0
2+80+10 =92

7.Divide the result by 19
find the remainder
mod(92/19)=16
If the remainder is zero (0) then the number is divisible by 19 else the result is remainder.

The remainder is 16. 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 19 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 19.
Remember the sequence 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2, 1, -9, 5, -7, 6,....
Here 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2 repeat in the sequence. Here 1, -9, 5, -7, 6, 3, -8, -4, -2 is followed by negative of them in order.
0.Reverse the order of
digits
6 7 8 3 5 6 2 4
1.Multiply with
the sequence
6×1 7×-9 8×5 3×-7 5×6 6×3 2×-8 4×-4
-- 6 -63 40 -21 30 18 -16 -16
Remainder= mod(-22/19) =-3
Real Remainder = 19-3 = 16
If the remainder is zero (0) then the number is divisible by 19 else the result is remainder.

The remainder is 16. Repeat the step 0,1 and 2 till you get a number whose remainder you can find easily.

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 <19 and >0 then the number is the remainder else 19+(negative number) is the remainder. -18≤negative number≤-1.