BananaLearnLearning LibraryGamesMake the greatest number of identical packs from 24 pencils and 36 stickers.
- Find HCF(24, 36) = 12.
- 24 ÷ 12 = 2 pencils; 36 ÷ 12 = 3 stickers per pack.
- Check 12 × 2 = 24 and 12 × 3 = 36. No leftovers.
Which question is the HCF answering?
The word greatest is not enough to select HCF. Read what must be equal and what must divide exactly. For gift packs, every pack must have the same number of pencils and the same number of stickers, and all items must be used. The number of packs must therefore divide both starting totals.
With 24 pencils and 36 stickers, one pack is possible but not the greatest number. Two packs contain 12 pencils and 18 stickers each. Twelve packs contain 2 pencils and 3 stickers each. Twenty-four packs fail because 36 stickers cannot be shared equally into 24 whole-item packs.
| Possible packs | Pencils per pack | Stickers per pack | Works with no leftovers? |
|---|---|---|---|
| 3 | 8 | 12 | Yes |
| 6 | 4 | 6 | Yes |
| 12 | 2 | 3 | Yes |
| 18 | 24 ÷ 18 | 2 | No |
| 24 | 1 | 36 ÷ 24 | No |
Use a factor list when the numbers are small
List pairs in order so factors are not missed: for 24 use 1 × 24, 2 × 12, 3 × 8 and 4 × 6. For 36 use 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Common factors are 1, 2, 3, 4, 6 and 12; choose 12.
HCF identifies a common divisor, not a common multiple. If bells ring every 6 and 8 minutes and you need the next time they coincide, you need a time divisible by both intervals: that is an LCM problem. If 6 red and 8 blue beads must form identical packs, you need a divisor of both counts: that is HCF.
Use prime factors or repeated division for larger numbers
For 84 and 126, write 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7. Pair the shared factors once each: 2 × 3 × 7 = 42. Extra copies of 2 or 3 that appear in only one number cannot enter the common product.
Repeated division reaches the same result: 126 = 1 × 84 + 42, then 84 = 2 × 42 + 0. The last non-zero remainder is 42. Why does this work? Any number dividing both 126 and 84 also divides their difference, 42. The smaller pair retains the common divisors.
For three collections, all three must work
Suppose there are 30 pencils, 45 stickers and 60 cards. HCF(30,45) = 15, and 15 also divides 60. Fifteen packs contain 2 pencils, 3 stickers and 4 cards each. A divisor shared by only two totals is insufficient.
For ribbon cutting, the output changes units: the HCF is a piece length, not a number of packs. State the length first, then divide each ribbon by that length to find the number of pieces. Distinguishing the two answers is part of solving the problem.
Worked examples: follow each decision
Cut ribbons of 42 cm and 56 cm into the longest equal pieces with no waste.
- 42 = 2 × 3 × 7; 56 = 2³ × 7. Common product: 2 × 7 = 14.
- Each piece is 14 cm long. There are 3 pieces from one ribbon and 4 from the other.
- Seven pieces in total is different from the 14 cm piece length.
84 red, 126 blue and 210 green beads must form the greatest number of identical kits.
- HCF(84,126)=42. Check 210 ÷ 42 = 5.
- Make 42 kits, each with 2 red, 3 blue and 5 green beads.
- Check each total separately. Saying “42 beads” would label the answer incorrectly.
13 questions with explained answers
Attempt a short group of questions before opening the solutions. Use a drawing or written method where helpful. Explain the answer, then try a similar question with a detail changed.
HCF of 18 and 30
Show answer and reasoning
6
18 = 2 × 3² and 30 = 2 × 3 × 5; common product is 6.
Greatest number of identical packs from 20 pens and 28 erasers
Show answer and reasoning
4 packs
Each pack contains 5 pens and 7 erasers.
Longest equal piece from 45 cm and 60 cm
Show answer and reasoning
15 cm
45 ÷ 15 = 3 and 60 ÷ 15 = 4.
HCF of 8 and 15
Show answer and reasoning
1
They have no common prime factor.
Is 24 the HCF of 24 and 36?
Show answer and reasoning
No; it is 12
24 does not divide 36 exactly.
Greatest number of identical packs using 18 pens and 24 cards?
Show answer and reasoning
6 packs; 3 pens and 4 cards each
HCF(18,24)=6. Both contents must be stated.
Longest equal pieces from 72 cm and 90 cm ribbons?
Show answer and reasoning
18 cm
72/18=4 and 90/18=5; nine pieces altogether.
HCF of 30, 45 and 75?
Show answer and reasoning
15
15 divides all three, and no larger common factor does.
Two bells ring every 8 and 12 minutes. HCF or LCM for their next shared ring?
Show answer and reasoning
LCM
A future time must be a multiple of both intervals; the first is 24 minutes.
Find HCF(84,126) using division.
Show answer and reasoning
42
126−84=42 and 84 is exactly divisible by 42.
Can 12 be the HCF of 48 and 60?
Show answer and reasoning
Yes
48=4×12 and 60=5×12; 4 and 5 have no further common factor.
Use 40 apples and 56 oranges in 8 identical baskets. Is 8 the greatest possible number?
Show answer and reasoning
Yes; HCF is 8
Each basket has 5 apples and 7 oranges.
HCF of two different prime numbers?
Show answer and reasoning
1
Their only shared positive factor is 1.
Check what you can now explain
- What must divide every total?
- Does your proposed factor work for all collections?
- Have you labelled the result as packs, length or contents per pack?
Keep these distinctions clear
- HCF makes exact groups from existing quantities; LCM finds a shared multiple. Do not choose by a keyword alone.
- Check that the proposed factor divides every count.
Save a useful copy
Download the practice and answer pack with BananaLearn branding and the matching guide address. Questions and answers are separated so you can attempt the work first.
Download printable PDFFor the complete explanation, use your browser’s Print → Save as PDF, or copy the guide link to share.
Keep the ideas connected
- Grade 6 Prime Time: factors, multiples and prime numbers
- Grade 5 Lowest Common Multiple: Lists, Cycles and Problems
- Choose a Maths practice game by skill
For help with an ongoing school-learning difficulty, read how Maths tutoring works or ask about English or Science support. Share a specific example your child found difficult.
Sources and scope
Original explanations, examples, visual models and practice by BananaLearn. This guide is not an official textbook extract, exam paper or endorsed assessment resource.
The examples teach the stated concept; check your school’s terminology and sequence when using the guide alongside a course.