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.