21

Divisibility by 21

Method 1:

The number is divisible by 21 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 21.
Remember the sequence 1, 10, 16, 13, 4, 19, 1, 10, 16, 13, 4, 19, 1, 10, 16, 13, 4, 19, 1, 10, 16,....
Here 1, 10, 16, 13, 4, 19 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 8×16 3×13 5×4 6×19 2×1 4×10
-- 6 70 128 39 20 114 2 40
2.Add the numbers 6+70+128+39+20+114+2+40=419
3.Repeat step 0 with
the added number
9 1 4 0 0 0 0 0
4.Multiply with
the sequence
9×1 1×10 4×16 0 0 0 0 0
-- 9 10 64 0 0 0 0 0
5.Add all
9+10+64 = 83
6.Divide the result by21
find the remainder
mod(83/21)=20
If the remainder is zero (0) then the number is divisible by 21 else the result is remainder.

The remainder is 20.  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 21 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 21.
Remember the sequence 1, 10, -5, -8, 4, -2, 1, 10, -5, -8, 4, -2, 1, 10, -5, -8, 4, -2, 1, 10, -5,...
Here 1, 10, -5, -8, 4, -2 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×-5 3×-8 5×4 6×-2 2×1 4×10
-- 6 70 -40 -24 20 -12 2 40
2.Add the numbers 6+70-40-24+20-12+2+40=62
3.Repeat step 0,1 and 2 with
the result, if number is very large and positive
Remainder is mod(62/21)=20
If the remainder is zero (0) then the number is divisible by 21.

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 21+negative value is the remainder. Suppose if we get -13 as sum then 21-13=8 is the remainder.

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