Divisibility Rules (Part-3)

In Part-1 and Part-2 we saw rules for divisibility from 2 to 50. Here in Part-3 we will see some more patterns of divisibility rules:

Rules for all divisors ending in 1...

Divisor	Multiply last digit by…
11	1 (begin with 1)
21	2
31	3
41	4
51	5 (add 1 each time)
and so on…

For example, to tell if a number is divisible by 31, multiply the last digit of a number by 3 and subtract it from the rest of the number.

For example, let’s see whether 34379 is divisible by 31.

3437 - 9×3 = 3410 which is also divisible by 31, therefore 34379 is divisible by 31.

Rules for all divisors ending in 3

Divisor	Multiply last digit by…
3	2 (begin with 2)
13	9
23	16
33	23
43	30
53	37 (add 7 each time)
and so on…

Rules for all divisors ending in 7

Divisor	Multiply last digit by...
7	2 (begin with 2)
17	5
27	8
37	11
47	14
57	17 (add 3 each time)
and so on...

Rules for all divisors ending in 9

Divisor	Multiply last digit by...
9	8 (begin with 2)
19	17
29	26
39	35
49	44
59	53 (add 9 each time)
and so on...

Rules for 2ⁿ or 5ⁿ

To see if a number is divisible by 2ⁿ or 5ⁿ; check to see if the last n digits are divisible by 2ⁿ or 5ⁿ. For example, 123489012349375 is divisible by 5³ because the last 3 digits are divisible by 125.

You can also combine division rules for composite numbers. For example, to tell if a number is divisible by 3298 (2×17×97) it must be divisible by 2, 17 and 97 which you can now figure out the rules for.

Using these rules and patterns in different combinations, you can figure the divisibility rules for absolutely any integer divisor.