Multiples , the times tables

If a tailor stitches 8 quilt patches per packet, then 1 packet has 8 patches, 2 packets have 16, 3 packets have 24, and so on. The numbers 8, 16, 24, 32, 40… are the multiples of 8.

What is this?

A multiple of a number is what you get when you multiply it by 1, 2, 3, 4… and so on. The multiples of any number are exactly its times table.

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, …

Multiples never end. Every number has infinite multiples.

Multiples and factors are flip sides of the same coin. If 3 is a factor of 12, then 12 is a multiple of 3. Read each direction:

• "3 is a factor of 12" → 3 divides 12 exactly.
• "12 is a multiple of 3" → 12 = 3 × 4.

Common multiples of two numbers are multiples that appear in both lists. For 4 and 6:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …
• Multiples of 6: 6, 12, 18, 24, 30, 36, …
• Common multiples: 12, 24, 36, … The smallest common multiple is called the LCM (Least Common Multiple). The LCM of 4 and 6 is 12.

Common factors are factors that appear in both lists. For 12 and 18:

• Factors of 12: 1, 2, 3, 4, 6, 12.
• Factors of 18: 1, 2, 3, 6, 9, 18.
• Common: 1, 2, 3, 6. The largest is called the HCF (Highest Common Factor) = 6.

Worked examples

Example 1. List the first 5 multiples of 7.

• 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, 7 × 5 = 35.
• Multiples: 7, 14, 21, 28, 35.

Example 2. Find the LCM of 3 and 5.

• Multiples of 3: 3, 6, 9, 12, 15, …
• Multiples of 5: 5, 10, 15, …
• LCM = 15.

Example 3. Find the HCF of 8 and 12.

• Factors of 8: 1, 2, 4, 8.
• Factors of 12: 1, 2, 3, 4, 6, 12.
• Common: 1, 2, 4. HCF = 4.

Try it

1. List the first 6 multiples of 9.
2. Find the LCM of 4 and 5.
3. Find the LCM of 6 and 8.
4. Find the HCF of 15 and 25.
5. Is 56 a multiple of 7? Why?

Hands-on

Hop counting. On a long row of stones, hop in 3s. Mark every 3rd stone with a leaf. Now hop again in 4s and mark every 4th stone with a flower. The stones with both marks are common multiples of 3 and 4.

Times-table chart. Write a 10 × 10 multiplication table. Pick any two numbers (say 6 and 8). Highlight all their multiples up to 60. The first one in both colours is the LCM.