### 23

Divisibility by 23

Method 1:

The number is divisible by 23 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 23.
Remember the sequence 1,10, 8, 11, 18, 19, 6, 14, 2, 20, 16, 22, 13, 15, 12, 5, 4, 17, 9, 21, 3, 7, 1, 10, 8, 11, 18, 19, 6 ...
Here 1,10, 8, 11, 18, 19, 6, 14, 2, 20, 16, 22, 13, 15, 12, 5, 4, 17, 9, 21, 3, 7  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 8×8 3×11 5×18 6×19 2×6 4×14
-- 6 70 64 33 90 114 12 56
2.Add the numbers 6+70+64+33+90+114+12+56 = 445
3.Repeat step 0 with
5 4 4 0 0 0 0 0
4.Multiply with
the sequence
5×1 4×10 4×8 0 0 0 0 0
-- 5 40 32 0 0 0 0 0
5+40+32 =77
6.Divide the result by 23
find the remainder
mod(77/23)=8
If the remainder is zero (0) then the number is divisible by 23 else the result is remainder.

The remainder is 8.  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 23 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 23.
Remember the sequence 1, 10, 8, 11, -5, -4, 6, -9, 2, -3, -7, -1, -10, -8, -11, 5, 4, -6, 9, -2, 3, 7, 1, 10, 8, 11, -5, -4, 6, ...
Here 1, 10, 8, 11, -5, -4, 6, -9, 2, -3, -7, -1, -10, -8, -11, 5, 4, -6, 9, -2, 3, 7 repeat in the sequence. 1, 10, 8, 11, -5, -4, 6, -9, 2, -3, -7 followed by their negatives in order.
0.Reverse the order of
the digits
6 7 8 3 5 6 2 4
1.Multiply with
the sequence
6×1 7×10 8×8 3×11 5×-5 6×-4 2×6 4×-9
-- 6 70 64 33 -25 -24 12 -36
3.Repeat step 0,1 and 2 with
the result, if number is very large and positive
Remainder is mod(100/23)=8
If the remainder is zero (0) then the number is divisible by 23.

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 for the remainder. If the result is positive then its remainder is the remainder if the result is negative then 23+negative value is the remainder. Suppose if we get -14 as sum then 23-14=9 is the remainder.