An expert question on remainders for P5–P6, 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
Primary and secondary pupils from all over Hong Kong join a mass display; the number of pupils is a four-digit number.
Lined up in groups of 13, the last group is 5 short; in groups of 15, 6 are left over; and with twice as many (everyone bringing a friend) in groups of 11, 4 are left over.
If as many pupils as possible take part, how many is that?
Show the hints (3)
💡Hint 1: Twice the number leaves 4 on division by 11. If the number itself leaves x, then 2 × x leaves 4 too; 2 × x is less than 22, so what can it be? "The last group is 5 short": what remainder on division by 13 is that? Then take two conditions and find the smallest number meeting both.
💡Hint 2: Find the smallest number fitting the groups of 15 and of 13; it comes round every 15 × 13 = 195. Among those, find one that fits the groups of 11.
💡Hint 3: The smallest number fitting the groups of 15 and of 13 is 21; it comes round every 195.
Worked solution
Have a go first, then read the solution step by step.
1.Twice the number leaves 4 on division by 11: if the number leaves x (less than 11), 2 × x leaves 4 too. 2 × x is less than 22, so it is 4 or 4 + 11 = 15; the even one is 4, so x = 2: the number leaves 2 on division by 11.
2.The three conditions: leaving 8 when divided by 13; leaving 6 when divided by 15; leaving 2 when divided by 11.
3.Numbers leaving 6 when divided by 15: 6, 21, … The first that leaves 8 when divided by 13 is 21.
4.Numbers meeting both come every 15 × 13 = 195: 21, 216, 411, 606, 801, …, 1581, 1776, 1971. The first that leaves 2 when divided by 11 is 1971 (the 11th).
5.13, 15, 11 share no factors, so numbers meeting all three come every 13 × 15 × 11 = 2145: 1971, 4116, 6261, …
6.The largest four-digit number is 9999. (9999 − 1971) ÷ 2145 = 3 remainder 1593, so the largest is 1971 + 3 × 2145 = 8406.
⚠️Common mistake: halving 4 for the number itself, forgetting 2 × x can be 15; or stopping at the smallest, 1971. And "5 short" means a remainder of 8 on division by 13, not 5.
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