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Maths olympiad question · P6★★★★ Expert

Divisibility and factors: olympiad question 9

An expert question on divisibility and factors for P6, with a step-by-step solution and the answer. Common in the Hua Xia Cup's P4 and P5 qualifying rounds and finals; harder ones use how many factors a number has, and square numbers.

The question

A maths-camp guessing game: the tutor thinks of two different whole numbers and gives two clues:

(1) they add up to 825;

(2) their lowest common multiple is exactly 156 times their highest common factor.

The pupils find more than one pair fits.

Add up the larger number of every pair that fits. What is the total?

Show the hints (3)
  1. Hint 1: Let the HCF be g, so the numbers are g × p and g × q with p and q sharing no factor. What is the LCM?
  2. Hint 2: LCM = g × p × q, so p × q = 156; the sum g × (p + q) = 825, so p + q divides 825. Split 156 into two coprime factors and try each.
  3. Hint 3: 156 splits as 1 × 156, 2 × 78, 3 × 52, 4 × 39, 6 × 26, 12 × 13. First drop the ones that aren't coprime.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Let the HCF be g and the numbers g × p and g × q (p < q, coprime — else the HCF would be bigger). The LCM is g × p × q, so p × q = 156; the sum g × (p + q) = 825, so p + q divides 825.
  2. 2.1 × 156: 1 + 156 = 157, which doesn't divide 825 — no
  3. 3.2 × 78: 2 and 78 share 2 — no
  4. 4.3 × 52: 3 + 52 = 55, HCF = 825 ÷ 55 = 15, the numbers 45 and 780 ✓
  5. 5.4 × 39: 4 + 39 = 43, which doesn't divide 825 — no
  6. 6.6 × 26: 6 and 26 share 2 — no
  7. 7.12 × 13: 12 + 13 = 25, HCF = 825 ÷ 25 = 33, the numbers 396 and 429 ✓
  8. 8.So the larger numbers are 780, 429, adding to 780 + 429 = 1209.
  9. 9.Check (first pair): 45 + 780 = 825, HCF 15, LCM 2340 = 156 × 15 ✓
  10. ⚠️Common mistake: using every split of 156 and forgetting p and q must be coprime (e.g. 2 × 78 won't do); or stopping at the first pair.

Answer: 1209

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