Olympiad topic · P4–P6
Divisibility and factors
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.
Try one
★★ MediumA florist has 60 red roses and 108 white roses.
She wants to tie all of them into identical bunches: every bunch has the same number of red roses and the same number of white roses, with none left over.
If she makes as many bunches as possible, how many roses are in each bunch?
- Hint 1: The number of bunches must share out both the 60 red and the 108 white roses exactly.
- Hint 2: First find the largest number that divides both 60 and 108: that is the most bunches.
- 1.Both colours are cut into the same number of bunches, all alike, so the number of bunches must divide both 60 and 108.
- 2.Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60; factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. The blue ones are common factors.
- 3.The largest common factor is 12, so at most 12 bunches.
- 4.Each bunch: 60 ÷ 12 = 5 red and 108 ÷ 12 = 9 white, 5 + 9 = 14 roses.
- ⚠️Common mistake: answering 12, the number of bunches, not the roses in each.
Answer: 14 roses
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsCounting multiples, divisibility tests.
- ★★MediumFree3 question types · unlimited questionsEqual groups, things happening together, factor pairs.
- ★★★HardFree2 question types · unlimited questionsThe lockers puzzle (number of factors), smudged digits.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsA set number of factors, HCF and LCM together, perfect squares.
Divisibility and factors: what it is and how to solve it
Whether one number divides another, and factors, multiples, highest common factors and lowest common multiples.
For example: A book number 3□85 divides by 3. The largest □? The digits add to 16 + □, a multiple of 3, so □ is 2, 5 or 8: the largest is 8.
The key idea: Know the tests: 2 and 5 by the last digit, 4 by the last two, 3 and 9 by the digit sum. Sharing into equal groups or two things happening together usually means the HCF or LCM. The usual method finds them by listing or by short division — the same idea.
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