Start with factors and multiples
A factor divides a whole number without a remainder. For example, 7 is a factor of 42 because 42 = 7 × 6. A positive multiple is made by multiplying a number by a positive whole number: 42, 84 and 126 are positive multiples of 42. A common factor divides both numbers; a common multiple is a multiple of both.
| Question about positive whole numbers | Object you are looking for | Selection |
|---|---|---|
| Which numbers divide both 42 and 56? | Common factors: 1, 2, 7, 14 | Greatest is 14: the GCF |
| Which positive numbers are multiples of both 8 and 12? | Common multiples: 24, 48, 72, … | Least positive is 24: the LCM |
Example 1: make the largest number of identical kits
You have 42 pencils and 56 stickers. Make identical kits using every item, with the same number of pencils in each kit and the same number of stickers in each kit. What is the largest possible number of kits? The number of kits must divide both totals with no remainder. That makes it a common-factor question.
- 1
List the factors
42 has factors 1, 2, 3, 6, 7, 14, 21, 42. 56 has factors 1, 2, 4, 7, 8, 14, 28, 56.
- 2
Choose the greatest common one
The shared factors are 1, 2, 7 and 14. The largest is 14, so you can make 14 kits.
- 3
Check the meaning
Each kit has 42 ÷ 14 = 3 pencils and 56 ÷ 14 = 4 stickers. Multiplying back gives exactly 42 pencils and 56 stickers.
The answer 14 counts kits, not items per kit. A question asking for the largest number of pencils in one kit could have a different answer or need more conditions. Naming the unknown and its unit prevents the correct calculation from being attached to the wrong quantity.
Greatest common factorPractise GCF calculations, then explain which quantity the factor would represent in a grouping problem.Example 2: when do two signals coincide again?
Two signals flash together now. One repeats every 8 minutes and the other every 12 minutes. How many minutes pass before they next flash together? The waiting time must be a positive multiple of both intervals. You want the earliest positive time, so this is a least-common-multiple question.
| Signal | Positive elapsed times in minutes |
|---|---|
| Every 8 minutes | 8, 16, 24, 32, 40, 48, … |
| Every 12 minutes | 12, 24, 36, 48, … |
The first shared positive time is 24 minutes. Check: 24 ÷ 8 = 3 intervals and 24 ÷ 12 = 2 intervals. Zero is a shared starting time, but it is not the next event. If the signals start at different times, or do not repeat steadily, this direct LCM model does not apply without additional reasoning.
Least common multiplePractise finding the first shared positive multiple; the generated exercises are numerical, not automatically written signal stories.Do not choose from a single keyword
“Least” does not always mean LCM. The least possible number of identical kits in the pencil example is one kit containing everything. Likewise, “greatest” does not always mean GCF. Draw or write the relationship first: does the unknown divide both totals, or is it made from whole repeats of each given interval? Then apply any maximum or minimum condition.
Try two new problems
A teacher uses all 30 blue counters and 45 red counters to make the greatest possible number of identical bags. A second pair of signals repeats every 6 and 10 minutes and starts together. Find the number of bags and the next shared flashing time, and state what each answer measures.
Separate modelling from calculation
If the relationship is clear but listing factors is difficult, use factor-pair practice. If multiples are being confused with factors, write the first few products alongside the divisor list for one number. Then mix a few GCF and LCM questions and ask learners to explain each term. A mixed numerical sheet does not assess whether a learner can interpret every word problem.
Factor pairsBuild complete divisor lists without skipping a matching factor.Finding multiplesPractise repeated products before comparing two lists.