Multiples of a multiple

The idea

If A is divisible by k, then all multiples of A are divisible by k.

Packets, boxes and orders

A packet holds 7 sweets. A box holds 2 packets, so 14 sweets. A shop orders 5 boxes: 14 × 5 = 70 sweets.

Can all 70 sweets be put back into packets of 7, with none left over?

• Yes, none left over
• No, some are left over
• It depends on how many boxes are ordered

Make copies

A is built from blocks of 7. Change the blocks in A and the number of copies. Every copy is a group of blocks. What is every total made of?

Copies of multiples

If A is divisible by k, then all multiples of A are divisible by k.

A = k × j, so A × m = (k × j) × m = k × (j × m). It is k times a whole number.

Examples: 28 = (7 × 2) × 2 70 = (7 × 2) × 5 154 = (7 × 2) × 11

Notes

If A is divisible by k, then all multiples of A are divisible by k.

Check yourself

56 is divisible by 7. Which of these must also be divisible by 7?

91 is divisible by 13 (91 = 13 × 7). What is 91 × 6 ÷ 13?

Answer: 42

91 × 6 = (13 × 7) × 6 = 13 × 42 = 546. So 546 ÷ 13 = 42.

6 is not divisible by 4. Which multiple of 6 is not divisible by 4?

27 is divisible by 9. How many of its multiples 27, 54, 81, 108, 135 are divisible by 9?

Answer: 5

27 = 9 × 3, so every multiple 27 × m = 9 × (3 × m) is divisible by 9: 54 = 9 × 6, 81 = 9 × 9, 108 = 9 × 12, 135 = 9 × 15. All 5.

• 56 + 5. 56 + 5 = 61 leaves remainder 5. A multiple of 56 is 56 × m, not 56 + 5.
• 56 × 9 — correct. Yes! 56 × 9 = 504 = 7 × 72. A multiple of a multiple of 7 is a multiple of 7.
• 56 − 3. 56 − 3 = 53 leaves remainder 4. A multiple of 56 is 56 × m, not 56 − 3.
• 56 + 1. 56 + 1 = 57 is one more than a multiple of 7. A multiple of A means A × m, not A + 1.
• 12. 12 = 4 × 3 is divisible by 4.
• 24. 24 = 4 × 6 is divisible by 4.
• 30 — correct. Yes! 30 = 4 × 7 + 2. The rule needs A itself to be divisible by 4. 6 is not, so some multiples of 6 are not.
• 36. 36 = 4 × 9 is divisible by 4.