A hard question on divisibility and factors for P5–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
The school ordered a batch of identical T-shirts for the choir at $72 each.
The total on the delivery note is a four-digit number, but rain has blurred its first and last digits: all you can read is "$□94□".
What is the total in dollars?
Show the hints (3)
💡Hint 1: The total is a multiple of 72. Which two easy-to-check numbers multiply to 72?
💡Hint 2: Use divisibility by 8 first to find what the last digit can be.
💡Hint 3: Then use divisibility by 9 (the digit sum) to find the first digit.
Worked solution
Have a go first, then read the solution step by step.
1.72 = 8 × 9, and 8 and 9 share no factor above 1, so the total must be divisible by both 8 and 9.
2.For 8, only the last three digits "94□" matter: 944 ÷ 8 = 118, so the last digit must be 4.
3.Last digit 4: the digit sum □ + 9 + 4 + 4 = □ + 17 must be a multiple of 9, so □ = 1.
4.The total is $1944. Check: 1944 ÷ 72 = 27, so 27 T-shirts were bought.
⚠️Common mistake: thinking divisibility by 8 just needs an even last digit, or that being divisible by 4 and 18 is enough for 72 (they share the factor 2).
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