Recall that a shortcut to check whether a given number is divisible by 3 is to add the digits of the number and check if the sum is a multiple of 3. Express the relationship between ‘a number is divisible by 3’ and ‘sum of the digits is a multiple of 3’ using ‘If-then’ sentences.

Step-by-step solution

Idea: A number and the sum of its digits always differ by a multiple of 3, because 10, 100, 1000, … are each 1 more than a multiple of 3 (9, 99, 999, …). So if either one is a multiple of 3, so is the other.

1. Proposition: If a number is divisible by 3, then the sum of its digits is a multiple of 3.½ mark
2. Converse: If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3. (This is the direction the shortcut uses.)½ mark
3. Why both hold: 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1, … So a number minus its digit sum is a multiple of 9, hence of 3. E.g. 4521 − (4 + 5 + 2 + 1) = 4 × 999 + 5 × 99 + 2 × 9 = 4509 = 3 × 1503.1 mark
4. The number and its digit sum differ by a multiple of 3, so if one is a multiple of 3, the other is too. Both ‘If-then’ sentences are true.½ mark
5. Together: a number is divisible by 3 if and only if the sum of its digits is a multiple of 3. E.g. 4521: 4 + 5 + 2 + 1 = 12, and 4521 = 3 × 1507.½ mark

Check: 4522: digit sum 13, not a multiple of 3, and 4522 = 3 × 1507 + 1 is not divisible by 3 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing only one ‘If-then’ sentence. The relationship has two directions, and the shortcut uses the second one (digit sum ⇒ divisible).
• Writing ‘If a number is not divisible by 3, then …’ — that is not the converse.
• Applying the same shortcut to 6 or 4. Adding digits works for 3 and 9 because 10 is one more than a multiple of 3 and of 9.

How this can come in the exam

Try one yourself

Without dividing, decide whether 98,765 is divisible by 3.

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