### 22

Divisibility by 22

Method 1:

The number is divisible by 22 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 22.
Remember the sequence 1, 10, 12, 10, 12, 10, 12, 10, 12, .....
Here 10, 12 repeat in the sequence. 1 is present at the beginning only.
0.Reverse the order of
the digits
6 7 8 3 5 6 2 4
1.Multiply with
the sequence
6×1 7×10 8×12 3×10 5×12 6×10 2×12 4×10
-- 6 70 96 30 60 60 24 40
2.Add the numbers 6+70+96+30+60+60+24+40 = 386
3.Repeat step 0 with
6 8 3 0 0 0 0 0
4.Multiply with
the sequence
6×1 8×10 3×12 0 0 0 0 0
-- 6 80 36 0 0 0 0 0
6+80+36 = 122
6.Divide the result by 22
find the remainder
mod(122/22)=12
If the remainder is zero (0) then the number is divisible by 22 else the result is remainder.

The remainder is 22.  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 22 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 22.
Remember the sequence  1, 10, -10, 10, -10, 10, -10, 10, -10 .....
Here 10, -10 repeat in the sequence. 1 is present just at the beginning.
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×-10 3×10 5×-10 6×10 2×-10 4×10
-- 6 70 -80 30 -50 60 -20 40
2.Add the numbers 6+70-80+30-50+60-20+40=56
3.Repeat step 0,1 and 2 with
the result, if number is very large and positive
Remainder is mod(56/22)=12
If the remainder is zero (0) then the number is divisible by 22.

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 22+negative value is the remainder. Suppose if we get -14 as sum then 22-14=8 is the remainder.