Olympiad topic · P3–P6
Remainders
Common in the Hua Xia Cup's P3 to P6 papers; harder ones need several remainder conditions to hold at once.
Try one
★★ MediumMs Chan has 55 pencils.
She shares them equally among her class, and 1 is left over — not enough for one more each.
The class has between 19 and 32 pupils.
How many pupils are in the class?
- Hint 1: Without the 1 left over, the pencils would share out exactly. How many pencils is that?
- Hint 2: The class size is a factor of 55 − 1, between 19 and 32.
- 1.Pencils given out: 55 − 1 = 54, so the class size is a factor of 54.
- 2.Between 19 and 32, the only factor of 54 is 27 (54 = 27 × 2).
- 3.Check: 55 ÷ 27 = 2 remainder 1, and 1 is less than 27, as it should be.
- ⚠️Common mistake: looking for factors of 55 without first taking away the 1 left over.
Answer: 27 pupils
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsRounding up, the most there could be.
- ★★MediumFree2 question types · unlimited questionsLast digits that repeat, the divisor from remainders.
- ★★★HardFree4 question types · unlimited questionsSeveral conditions at once, three batches with equal remainders, too few left to share again.
- ★★★★Expert (competition level)VIP4 question types · unlimited questionsHow many four-digit numbers fit, the largest or nearest four-digit one, the last digit of 1¹ + 2² + 3³ + …, numbers whose quotient equals the remainder.
Remainders: what it is and how to solve it
What's left when a division isn't exact is the remainder; these problems use how remainders behave.
For example: 35 pupils, at most 8 to a table. How many tables at least? 35 ÷ 8 = 4 r 3; the last 3 need a table too: 4 + 1 = 5.
The key idea: Dividend = divisor × quotient + remainder, and the remainder is always less than the divisor. With several conditions, try from the largest divisor; "short of a full box by 2" means adding 2 makes it divide exactly.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download