Easy divisibility rules that hardly anybody knows (Part 2)

The previous blog post described the “chop, subtract” rule for determining divisibility by 11. You chop the number’s ones digit and subtract it from what remains. If that difference is a multiple of 11 (0, 11, 22, 33, …), so is the original number. If that difference is not a multiple of 11, neither is the original number.

For example, 1083 is not a multiple of 11, because 108 – 3 = 105, and 105 is not a multiple of 11. On the other hand, 792 is a multiple of 11, because 79 – 2 = 77.

Rule for divisibility by 7

The rule for divisibility by 7 is, “chop, double, subtract.” This is like the rule for 11, but you double the chopped digit before subtracting.

EXAMPLE 1: Is 604 divisible by 7?
ANSWER 1: Chop the 4, double it (8) and subtract: 60 – 8 = 52. Because 52 is not a multiple of 7, neither is 604.

EXAMPLE 2: Is 1057 divisible by 7?
ANSWER 2: Chop the 7, double it (14) and subtract: 105 – 14 = 91. Because 91 is a multiple of 7, so is 1057. To verify that 91 is a multiple of 7, apply the rule again: chop the 1, double it (2), and subtract: 9 – 2 = 7.

Why divisibility rules work in general

The principle behind divisibility rules is that adding or subtracting multiples of n does not change whether a quantity is divisible by n. For example, 45 has a remainder of 3 when divided by 7. Adding multiples of 7 to 45 (52, 59, 66, 73 . . . ), produces numbers with remainders of 3. Similarly, subtracting multiples of 7 from 45 (38, 31, 24, 17 . . .), produces numbers with remainders of 3.

On the other hand, if a number is a multiple of, say 13, adding or subtracting 13 simply produces another multiple of 13.

Why the divisibility rules for 11 and 7 work

When you “chop, subtract,” you subtract a multiple of 11, because you subtract the number from both the ones place and the tens place (and 1+10 = 11). When you “chop, double, subtract,” you subtract a multiple of 21, because you subtract from the ones digit and TWICE from the tens digit (1+2×10 = 21). Every multiple of 21 is also a multiple of 7, so “chop, double, subtract” works as a divisibility rule for both 7 and 21.

PRACTICE: Use “chop, double, subtract” to prove that these are divisible by 21:    273    861    1092    27,531. Then prove that these are NOT divisible by 21:    281    854    1101    27,538.

PRACTICE: Use “chop, double, subtract” to prove that these are divisible by 7:    294    574    2233    29,449. Then prove that these numbers are NOT divisible by 7:    292    571    2244    29,452.