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
For the school open day, the parent–teacher association baked between 200 and 360 cookies.
Packed in boxes of 7, there are 2 left over; in boxes of 3, the last box is 1 short of full; and in boxes of 8, there are 5 left over.
How many cookies are there?
Show the hints (3)
💡Hint 1: "The last box is 1 short of full": so what is left when dividing by 3?
💡Hint 2: Find the smallest number that fits both the boxes of 8 and the boxes of 7; after that it comes round every 8 × 7 = 56.
💡Hint 3: Among those, find the one that also fits the boxes of 3 and lies between 200 and 360.
Worked solution
Have a go first, then read the solution step by step.
1.The three conditions: leaving 2 when divided by 7; leaving 2 when divided by 3; leaving 5 when divided by 8.
2.Numbers leaving 5 when divided by 8: 5, 13, 21, 29, 37, … The first that leaves 2 when divided by 7 is 37.
3.Numbers meeting both come every 8 × 7 = 56: 37, 93, 149, … The first that leaves 2 when divided by 3 is 149.
4.Numbers meeting all three come every 8 × 7 × 3 = 168: 149, 317, … The only one between 200 and 360 is 317.
⚠️Common mistake: reading "1 short of full" as "1 left over".
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