A medium question on divisibility and factors for P5, 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 hall needs 52 chairs set out in a rectangle: every row has the same number of chairs, and all the chairs are used.
There must be at least 2 rows and at least 2 chairs in each row.
"2 rows of 26" and "26 rows of 2" count as two different arrangements.
How many arrangements are there?
Show the hints (2)
💡Hint 1: Rows times chairs per row must make 52. Which numbers can the number of rows be?
💡Hint 2: List every factor of 52, then think about which numbers of rows aren't allowed.
Worked solution
Have a go first, then read the solution step by step.
1.The number of rows must be a factor of 52: 1, 2, 4, 13, 26, 52 — 6 of them. Each pair, top times bottom, makes 52: 2 rows means 26 in each.
2.1 row (of 52) is not allowed, and neither is 52 rows (of 1).
3.6 − 2 = 4.
⚠️Common mistake: treating 2 rows and 26 rows as the same and counting half, or forgetting to drop 1 row and 52 rows.
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