Doubling both numbers doubles the HCF

The idea

If both numbers are doubled, each gains an extra factor 2 in its prime factorisation, which joins the largest common subpart, so the HCF doubles.

Double both

The HCF of 6 and 9 is 3. Now double both numbers: 12 and 18.

What will the HCF of 12 and 18 be?

• 6, the HCF doubles
• 3, it stays the same
• 12

Double and predict

Each number is a row of prime beads. The ringed beads are the ones they share. Their product is the HCF. Predict the new HCF, then double both numbers.

One new 2 joins the shared beads

270 = 2 × 3 × 3 × 3 × 5 and 50 = 2 × 5 × 5. Shared beads: 2 × 5, so the HCF is 10.

Double both: 540 = 2 × 2 × 3 × 3 × 3 × 5 and 100 = 2 × 2 × 5 × 5. Each number got one extra 2. The two new 2s match, so they join the shared beads: 2 × 2 × 5, so the HCF is 20.

If both numbers are doubled, each gains an extra factor 2 in its prime factorisation. The two new 2s match and join the largest common subpart, so the HCF doubles.

• Numbers | Shared beads | HCF
• 270 and 50 | 2 × 5 | 10
• 540 and 100 | 2 × 2 × 5 | 20

Notes

If both numbers are doubled, each gains an extra factor 2, which joins the largest common subpart. So the HCF doubles: 270 and 50 have HCF 10, and 540 and 100 have HCF 20.

Check yourself

The HCF of 36 and 60 is 12. What is the HCF of 72 and 120, which are both doubled?

The HCF of 8 and 12 is 4. Both numbers are doubled three times: 16 and 24, then 32 and 48, then 64 and 96. What is the HCF of 64 and 96?

Answer: 32

Each doubling doubles the HCF: 4 → 8 → 16 → 32. So the HCF of 64 and 96 is 32.

The HCF of 6 and 9 is 3. Rani doubles only 6 to get 12 and 9. Is the HCF now 6?

• 12. Doubling both numbers does change the HCF. Each gains a 2, and the two 2s join the shared beads.
• 24 — correct. Yes! The HCF doubles: 12 × 2 = 24.
• 48. That is four times 12. Doubling once only doubles the HCF.
• Yes, the HCF doubles whenever one number is doubled.. Check it: 12 = 2 × 2 × 3 and 9 = 3 × 3. They share only 3, so the HCF is still 3.
• No, the HCF becomes 12.. 12 does not divide 9. The HCF of 12 and 9 is 3.
• No. The new 2 has no twin in 9, so it does not join the shared beads. — correct. Yes! Only when both numbers are doubled do the two new 2s match each other. The HCF of 12 and 9 is 3.