An expert question on divisibility and factors for P6, with a step-by-step solution, diagrams 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
Ms Chan buys a batch of prizes for an inter-class contest, and the number of prizes is a multiple of 10.
She notices there are exactly 8 different ways to split all the prizes into equal shares with nothing left over.
A different number of shares counts as a different way; keeping everything as 1 share, or giving 1 prize per share, each count as a way too.
What is the smallest number of prizes she could have?
Show the hints (3)
💡Hint 1: If 3 equal shares work with nothing left, how is 3 related to the number of prizes? So what do 8 ways tell you about its factors?
💡Hint 2: Write a number as a product of primes, like 72 = 2³ × 3²; its number of factors is (3 + 1) × (2 + 1) = 12.
💡Hint 3: Write 8 as a product: 8, 4 × 2, 2 × 2 × 2. Each way gives a pattern of prime powers, and the number must contain 10 = 2 × 5.
Worked solution
Have a go first, then read the solution step by step.
1.The number of shares in each way is a factor of the number of prizes, so that number has exactly 8 factors and is a multiple of 10.
2.If n = p^a × q^b × … (p, q, … different primes), n has (a + 1) × (b + 1) × … factors. For example 72 = 2³ × 3²: its factors fill 4 rows and 3 columns, (3 + 1) × (2 + 1) = 12 of them.
3.8 can be written as 8, 4 × 2, 2 × 2 × 2, so n can look like p⁷, p³ × q, p × q × r.
4.10 = 2 × 5, so n must have these primes, with powers no smaller. The smallest multiple of 10 of each pattern:
5.p⁷: impossible — it can't contain 2 × 5.
6.p³ × q: smallest is 2³ × 5 = 40.
7.p × q × r: smallest is 2 × 3 × 5 = 30.
8.The smallest is 30 = 2 × 3 × 5. Check: it has (1 + 1) × (1 + 1) × (1 + 1) = 8 factors.
⚠️Common mistake: answering 24 = 2³ × 3, which has 8 factors but is not a multiple of 10.
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