A hard question on remainders for P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 papers; harder ones need several remainder conditions to hold at once.
The question
A cha chaan teng gives 120 egg tarts and 78 pineapple buns to the elderly centres nearby.
The egg tarts are shared equally among the centres, and so are the buns.
12 egg tarts and 6 buns are left over — too few tarts to give every centre one more, and too few buns as well.
How many different numbers of centres are possible?
Show the hints (3)
💡Hint 1: Without the leftovers, both would share out exactly. How many tarts is that? How many buns?
💡Hint 2: The number of centres must divide both 108 and 72.
💡Hint 3: Too few to give one more each means the number of centres is more than 12 and more than 6.
Worked solution
Have a go first, then read the solution step by step.
1.Tarts handed out: 120 − 12 = 108; buns: 78 − 6 = 72. The number of centres is a common factor of 108 and 72.
2.108 = 36 × 3 and 72 = 36 × 2, with 3 and 2 sharing no factor above 1, so the HCF is 36 and the common factors are the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
3.12 egg tarts left is too few for one more each, so there are more than 12 centres: 18, 36 — 2 possibilities.
⚠️Common mistake: only requiring more than 6 and counting 9, 12 too — but then the 12 egg tarts left would give every centre one more.
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