11+ Factors, Multiples and Primes: Questions with Answers
Distinguish factors from multiples, find HCF and LCM, and solve original number puzzles with eight explained practice answers.
When a question mentions two numbers, multiplying them together is not automatically the right move. Factors describe exact divisions of a number. Multiples describe repeated whole-number groups. Knowing which relationship the problem needs is more useful than memorising the initials HCF and LCM on their own.
These original questions support preparation for exams taken in 2027. They are not official school papers. Topic coverage and question formats vary: check your target school’s current information. Try the questions on paper before opening each answer.
Factors and multiples answer different questions
We use positive whole-number factors here. A factor divides a number without a remainder. The factors of 28 are 1, 2, 4, 7, 14 and 28, because the pairs 1 × 28, 2 × 14 and 4 × 7 all make 28. Check the pairs in order so that you do not omit one.
A positive multiple of 7 is made by multiplying 7 by a positive whole number: 7, 14, 21, 28, 35 and so on. Zero is also a multiple in wider mathematical usage, but when finding a lowest common multiple we mean the smallest positive one. The word positive matters.
| Question | A useful test |
|---|---|
| Is 4 a factor of 28? | Yes: 28 ÷ 4 = 7 exactly. |
| Is 28 a multiple of 4? | Yes: 4 × 7 = 28. |
| Is 8 a factor of 28? | No: dividing leaves a remainder. |
Prime does not mean odd
A prime number is a whole number greater than one with exactly two positive factors: 1 and itself. Two is prime even though it is even. Nine is odd but not prime, because 3 × 3 = 9. One is not prime: it has only one positive factor.
To test a small number, look systematically for a factor pair other than 1 and the number. For 29, check 2, 3, 4 and 5. None divides it exactly. After that, any new factor pair would have to repeat a smaller partner you have already checked, since 6 × 6 exceeds 29. Therefore 29 is prime.
HCF: making the largest equal groups
The highest common factor is the greatest positive number that divides both values. Suppose you have 32 red counters and 48 blue counters. You want identical packs with none left over. Each pack must contain the same number of red counters as every other pack, and likewise for blue. The greatest possible number of packs is HCF(32, 48) = 16. Each has 2 red and 3 blue counters.
Be careful about what that 16 represents: it is the number of packs, not the number of counters inside each pack. A number can be calculated correctly and still be attached to the wrong label.
LCM: when repeating events next coincide
A light flashes every 6 seconds and another every 14 seconds. They flash together now. Multiples of 14 are 14, 28 and 42; 42 is the first that is also divisible by 6. They next coincide after 42 seconds. Multiplying 6 × 14 gives 84, a common multiple but not the lowest one.
Eight original questions
1. List every positive factor of 30.
Show answer and explanation
1, 2, 3, 5, 6, 10, 15, 30
The pairs are 1 × 30, 2 × 15, 3 × 10 and 5 × 6. Write the list in increasing order.
2. Which are prime: 1, 2, 21, 29, 39?
Show answer and explanation
2 and 29
1 has only one factor; 21 = 3 × 7 and 39 = 3 × 13. Only 2 and 29 have exactly two positive factors.
3. Find the highest common factor of 24 and 40.
Show answer and explanation
8
The common factors are 1, 2, 4 and 8. The greatest is 8, not 40, which does not divide 24.
4. Find the lowest common multiple of 9 and 12.
Show answer and explanation
36
Positive multiples of 12 are 12, 24, 36. Of these, 36 is the first divisible by 9.
5. There are 42 pencils and 70 stickers. Make the greatest number of identical packs using everything. How many packs, and what goes in each?
Show answer and explanation
14 packs; 3 pencils and 5 stickers each
HCF(42, 70) = 14. Divide each original quantity by 14 to find the contents.
6. Two bells ring every 8 and 10 minutes. They ring together at 09:00. When do they next ring together?
Show answer and explanation
09:40
The lowest common multiple is 40 minutes: 8 × 5 = 10 × 4. Add 40 minutes to the starting time.
7. A puzzle asks for a number greater than 30 and less than 60 that is a multiple of 8 and a factor of 96. Do these clues give a unique answer?
Show answer and explanation
32 or 48; the clues do not give a unique answer
The multiples of 8 in the range are 32, 40, 48 and 56. Both 32 and 48 divide 96, so this information actually has two solutions: 32 and 48. A single-answer question needs another clue.
8. Add the clue “it is also a multiple of 3” to question 7. What is the number now?
Show answer and explanation
48
Only 48 from the two surviving candidates is divisible by 3. Test every clue before deciding a number puzzle is finished.
Why checking a puzzle matters
Question 7 deliberately exposes an incomplete question: its wording asks for a number, and both 32 and 48 satisfy the conditions. If you found 32, your arithmetic was not wrong. The follow-up adds a condition to make a unique answer. When practising, distinguish an error in your method from a question that allows more than one result.
Factor knowledge helps with simplifying in our fraction, decimal and percentage conversion guide. For calculation conventions, try order of operations, or return to the maths hub.
Curriculum context: Department for Education maths programme. The examples are original and do not reproduce official exam questions. Answers checked 24 October 2026.
Ready to practise?
Choose original practice and use the answer explanations to decide what to work on next. These are not official school papers.
Keep reading
More guides relevant to this topic.