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Math · Number Sense

Factors, Multiples, Primes: Common Factors and Multiples

Find all the factors of a number, list multiples, tell prime from composite, use divisibility tests, and solve problems with common factors and multiples.

Grades 4–5 · 26 practice questions · Original material
Learning objectiveList all factors of a whole number up to 100 in pairs, list multiples, classify numbers as prime or composite, use divisibility tests for 2, 3, 4, 5, 9 and 10, and find common factors and common multiples to solve problems.

The idea

Factors and multiples are two sides of multiplication. Because 3 × 4 = 12, the numbers 3 and 4 are factors of 12, and 12 is a multiple of both 3 and 4. A factor divides a number with nothing left over. A multiple is what you get when you multiply a number by 1, 2, 3 and so on.

To find every factor of a number, work in pairs and stop when the pairs meet. For 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6. The next candidate, 5, does not divide 24, and 6 is already in the list, so you are done. The factors are 1, 2, 3, 4, 6, 8, 12 and 24. Writing the pairs keeps you from missing the large factors.

A prime number has exactly two factors: 1 and itself. A composite number has more than two. The number 1 has only one factor, so it is neither. The number 2 is the only even prime, because every other even number is also divisible by 2.

Divisibility tests save time. A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8; by 5 if it ends in 0 or 5; by 10 if it ends in 0; by 3 if its digits add to a multiple of 3; by 9 if its digits add to a multiple of 9; and by 4 if its last two digits make a multiple of 4.

A common factor is a factor of two numbers at once. A common multiple is a multiple of two numbers at once. Use common factors to make equal groups with nothing left over. Use common multiples to ask when two repeating events happen together.

Check your answer by multiplying back: if you say 7 is a factor of 56, then 7 × 8 should equal 56.

Worked examples

Factor pairs of 24 and common multiples of 4 and 6Top panel: the factor pairs of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6, giving eight factors. Bottom panel: multiples of 4 (4, 8, 12, 16, 20, 24) and multiples of 6 (6, 12, 18, 24) on one number line from 0 to 24. The first number in both lists is 12, then 24.Factors come in pairs. Multiples line up on a number line.Original schematic, drawn to the labelled values.1 · Factor pairs of 24 (each pair multiplies to 24)1 × 24factors 1 and 24= 242 × 12factors 2 and 12= 243 × 8factors 3 and 8= 244 × 6factors 4 and 6= 242 · Multiples of 4 and of 6 up to 2402468101214161820222448121620246121824multiples of 6Orange boxes: common multiples 12 and 24. The least is 12.blue = multiples of 4
Original schematic, drawn to the labelled values. Factors of 24 come in pairs that multiply to 24, so you can stop when the pairs meet. Multiples of two numbers line up on one number line, and the first place both land is the least common multiple.

Example 1 · All the factors of 30

List every factor of 30.

  1. Start with 1 × 30.
  2. 2 × 15 works because 30 ÷ 2 = 15. 3 × 10 works. 4 does not divide 30. 5 × 6 works.
  3. The next number, 6, is already in a pair, so stop. Put the factors in order.

1, 2, 3, 5, 6, 10, 15, 30

Example 2 · Prime or composite?

Is 51 prime or composite?

  1. 51 is odd and does not end in 0 or 5, so try 3.
  2. The digits add to 5 + 1 = 6, and 6 is a multiple of 3, so 51 is divisible by 3.
  3. 51 = 3 × 17, so it has factors besides 1 and 51.

Composite (51 = 3 × 17).

Example 3 · Common factors and the greatest one

Find the greatest common factor of 16 and 40.

  1. Factors of 16: 1, 2, 4, 8, 16.
  2. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
  3. Common factors: 1, 2, 4, 8. The greatest is 8.

8

Example 4 · Least common multiple

Two lights blink. One blinks every 6 seconds and the other every 8 seconds. They blink together now. When is the next time?

  1. Multiples of 6: 6, 12, 18, 24, 30.
  2. Multiples of 8: 8, 16, 24, 32.
  3. The first number in both lists is 24.

After 24 seconds.

Example 5 · Choose factors or multiples

Which kind of number answers each question? (a) How many equal teams can 20 players make? (b) When do two bus schedules match again?

  1. (a) Equal teams need a number that divides 20: a factor.
  2. (b) Repeating schedules meet at a number both times reach: a common multiple.

(a) a factor of 20; (b) a common multiple.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Stopping the factor list at the half-way point.

Why this fails: Every factor below the square root has a partner above it. Listing 1, 2, 3, 4, 6 for 12 forgets 12 itself. Work in pairs and write both numbers.

Mixing up factor and multiple.

Why this fails: Factors are smaller than or equal to the number and are few; multiples are equal to or bigger than the number and go on forever. 3 is a factor of 12, and 12 is a multiple of 3.

Calling every odd number prime.

Why this fails: 9, 15, 21, 25 and 27 are odd and composite. Test small primes such as 3, 5 and 7 before you decide.

Thinking 1 is prime or that 2 is not.

Why this fails: A prime needs exactly two factors. 1 has one, so it is not prime. 2 has two (1 and 2), so it is prime, and it is the only even one.

Choosing the greatest common factor when the problem needs a common multiple.

Why this fails: Equal groups use a common factor. Repeating events that meet again use a common multiple. Ask whether the answer should be smaller or larger than the numbers you started with.

Foundation

Foundation practice

Factors, multiples, prime and composite numbers.

List the factors

1. Which list shows all of the factors of 12?

  1. 1, 2, 3, 4, 6
  2. 2, 3, 4, 6
  3. 1, 2, 3, 4, 6, 8, 12
  4. 1, 2, 3, 4, 6, 12
Find a factor

2. Which number is a factor of 28?

  1. 7
  2. 6
  3. 8
  4. 9
First multiples

3. Which list shows the first five multiples of 6?

  1. 12, 18, 24, 30, 36
  2. 1, 2, 3, 6, 12
  3. 6, 12, 18, 24, 30
  4. 6, 16, 26, 36, 46
Factor or multiple?

4. 12 is a ______ of 3.

  1. factor
  2. prime number
  3. multiple
  4. product of 12 and 3
Spot the prime

5. Which number is prime?

  1. 29
  2. 21
  3. 27
  4. 33
Spot the composite

6. Which number is composite?

  1. 2
  2. 3
  3. 5
  4. 9
Is 1 prime?

7. Why is 1 not a prime number?

  1. It is an odd number, and primes must be even.
  2. It has only one factor, but a prime has exactly two factors.
  3. It is too small to be a number with factors.
  4. It can be divided by every number.
Find a multiple

8. Which number is a multiple of 8?

  1. 18
  2. 48
  3. 28
  4. 58

Practice

Practice practice

Divisibility tests, factor pairs and prime factorization.

Factor pairs

9. A teacher arranges 18 desks in equal rows. How many different factor pairs does 18 have?

  1. 2
  2. 6
  3. 9
  4. 3
Common factors

10. Which list shows the common factors of 12 and 18?

  1. 1, 2, 3, 6
  2. 1, 2, 3, 4, 6, 12
  3. 1, 2, 3, 6, 9, 18
  4. 6, 12, 18
Common multiple

11. What is the smallest number that is a multiple of both 4 and 6?

  1. 10
  2. 24
  3. 12
  4. 2
Divisibility by 3

12. Which number is divisible by 3?

  1. 124
  2. 140
  3. 135
  4. 251
Divisibility by 4

13. Which number is a multiple of 4?

  1. 316
  2. 322
  3. 330
  4. 342
Divisibility by 9

14. Which number is divisible by 9?

  1. 206
  2. 215
  3. 218
  4. 207
Prime factorization

15. Which is the prime factorization of 36?

  1. 4 × 9
  2. 2 × 2 × 3 × 3
  3. 6 × 6
  4. 2 × 18
Count the factors

16. How many factors does 30 have?

  1. 4
  2. 8
  3. 6
  4. 10
Missed factor

17. A student lists the factors of 20 as 1, 2, 4, 5, 10. What did the student forget?

  1. 3, because 3 is the next whole number.
  2. 8, because 8 is a multiple of 4.
  3. Nothing; a number is not one of its own factors.
  4. 20, because every number is a factor of itself.

Think

Think practice

Find the rule behind a number or test a claim.

Exactly three factors

18. Which number has exactly three factors?

  1. 6
  2. 7
  3. 9
  4. 8
True or false

19. Which statement is true?

  1. Every multiple of 3 is also a multiple of 6.
  2. Every multiple of 6 is also a multiple of 3.
  3. Every even number is a multiple of 4.
  4. Every odd number is prime.
Packs and boxes

20. Stickers come in packs of 3 and card holders come in packs of 5. What is the smallest number of stickers you can buy to have exactly the same number of stickers as card holders?

  1. 8
  2. 15
  3. 30
  4. 2
Factor trap

21. A rectangle has an area of 16 square units, and both side lengths are whole numbers. Which pair of side lengths is NOT possible?

  1. 3 and 5
  2. 1 and 16
  3. 2 and 8
  4. 4 and 4

Challenge

Challenge practice

Use factors and multiples in real situations.

Real-life multiples

22. Bus A leaves the stop every 4 minutes and Bus B every 6 minutes. They leave together at 3:00. When do they next leave together?

  1. 3:12
  2. 3:10
  3. 3:24
  4. 3:06
Mystery number

23. I am a prime number between 30 and 40. My ones digit is 7. What number am I?

  1. 33
  2. 35
  3. 37
  4. 39
Greatest common factor

24. What is the greatest common factor of 24 and 36?

  1. 12
  2. 6
  3. 72
  4. 4
Equal groups

25. A class has 24 students and 36 library books. The teacher makes identical groups with no books or students left over. What is the greatest number of groups she can make?

  1. 6
  2. 60
  3. 72
  4. 12
Error check

26. A student says: 'All prime numbers are odd.' Which number shows the student is wrong?

  1. 3
  2. 9
  3. 15
  4. 2

Answer key

  1. 1. 1, 2, 3, 4, 6, 12
    Factors are the whole numbers that divide 12 with no remainder: 1×12, 2×6 and 3×4 give 1, 2, 3, 4, 6, 12. Leaving out 1 and 12 forgets that every number is a factor of itself and has 1 as a factor. Adding 8 fails because 12 ÷ 8 leaves a remainder.
  2. 2. 7
    28 ÷ 7 = 4 with no remainder, so 7 is a factor. 28 ÷ 6, 28 ÷ 8 and 28 ÷ 9 all leave remainders. Notice that 8 and 9 sit near the factors 7 and 14 but do not divide 28.
  3. 3. 6, 12, 18, 24, 30
    Multiples of 6 come from skip counting by 6: 6, 12, 18, 24, 30. The second list skips the first multiple, 6 × 1. The third list gives factors of 6 and 12 instead of multiples. The last list adds 10 each time instead of 6.
  4. 4. multiple
    12 = 3 × 4, so 12 is in the 3 times table. That makes 12 a multiple of 3, and 3 is a factor of 12. A factor is the smaller number that divides evenly; a multiple is a result of multiplying.
  5. 5. 29
    29 has only two factors, 1 and 29. 21 = 3 × 7, 27 = 3 × 9 and 33 = 3 × 11, so all three have more than two factors. They look like primes because they are odd and not in the 2 times table.
  6. 6. 9
    9 = 3 × 3, so it has three factors (1, 3 and 9). A composite number has more than two factors. 2, 3 and 5 each have exactly two factors, so they are prime.
  7. 7. It has only one factor, but a prime has exactly two factors.
    A prime number has exactly two different factors: 1 and itself. The number 1 has just one factor, itself, so it is neither prime nor composite. Primes do not have to be even; 2 is the only even prime.
  8. 8. 48
    8 × 6 = 48, so 48 is a multiple of 8. 18, 28 and 58 all end in 8, which tricks the eye, but 18 ÷ 8, 28 ÷ 8 and 58 ÷ 8 leave remainders.
  9. 9. 3
    The factor pairs are 1 × 18, 2 × 9 and 3 × 6, so there are 3 pairs. The six factors are 1, 2, 3, 6, 9 and 18, but each pair uses two of them. Counting 6 mixes up the number of factors with the number of pairs.
  10. 10. 1, 2, 3, 6
    Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors appear in both lists: 1, 2, 3, 6. The second list is all the factors of 12 and the third list is all the factors of 18. The last list gives common multiples, not factors.
  11. 11. 12
    Multiples of 4: 4, 8, 12, 16… Multiples of 6: 6, 12, 18… The first one in both lists is 12. 24 is also a common multiple but not the smallest. 2 is a common factor, not a multiple.
  12. 12. 135
    Add the digits: 1 + 3 + 5 = 9, and 9 is in the 3 times table, so 135 is divisible by 3 (135 ÷ 3 = 45). The digits of 124 add to 7, of 140 to 5 and of 251 to 8, so none of them divide by 3.
  13. 13. 316
    Look at the last two digits. 16 is in the 4 times table (4 × 4), so 316 = 4 × 79. The last two digits 22, 30 and 42 are not multiples of 4. All four numbers are even, but being even is not enough for a multiple of 4.
  14. 14. 207
    The digits of 207 add to 2 + 0 + 7 = 9, so 207 is divisible by 9 (9 × 23 = 207). The digit sums of 206, 215 and 218 are 8, 8 and 11, so none of them divide by 9.
  15. 15. 2 × 2 × 3 × 3
    Break 36 down until every factor is prime: 36 = 4 × 9 = 2 × 2 × 3 × 3. The other choices are correct products of 36 but contain composite numbers (4, 9, 6 and 18), so they are not prime factorizations.
  16. 16. 8
    Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. That is 8. Listing only the numbers up to 6 gives 6 and forgets the partners 10, 15 and 30.
  17. 17. 20, because every number is a factor of itself.
    20 ÷ 20 = 1, so 20 is a factor of itself. The list has all the other factors correctly. 3 and 8 do not divide 20 evenly.
  18. 18. 9
    The factors of 9 are 1, 3 and 9, which is exactly three. 7 has two factors (it is prime); 6 has four (1, 2, 3, 6) and 8 has four (1, 2, 4, 8).
  19. 19. Every multiple of 6 is also a multiple of 3.
    6 = 3 × 2, so any multiple of 6 can be split into groups of 3. The reverse is false: 9 is a multiple of 3 but not of 6. 6 is even but not a multiple of 4, and 9 is odd but not prime.
  20. 20. 15
    The number must be a multiple of both 3 and 5, so find the least common multiple. Multiples of 3: 3, 6, 9, 12, 15. Multiples of 5: 5, 10, 15. The first match is 15. 30 also works but is not the smallest; 8 is the sum of 3 and 5; 2 is not a multiple of either.
  21. 21. 3 and 5
    The side lengths must be a factor pair of 16: 1 × 16, 2 × 8 or 4 × 4. 3 × 5 = 15, not 16, so it does not work.
  22. 22. 3:12
    Find the least common multiple of 4 and 6. Multiples of 4: 4, 8, 12. Multiples of 6: 6, 12. They match at 12 minutes, so 3:12. At 3:06 only Bus B leaves; 3:24 is later but not the first time.
  23. 23. 37
    The numbers between 30 and 40 with a 7 in the ones place are 37 only; 37 has just the factors 1 and 37, so it is prime. 33, 35 and 39 are composite, with 3 × 11, 5 × 7 and 3 × 13.
  24. 24. 12
    Common factors of 24 and 36: 1, 2, 3, 4, 6, 12. The greatest is 12. 6 is a common factor but not the greatest, and 72 is a common multiple, not a factor.
  25. 25. 12
    The number of groups must divide both 24 and 36, so it is a common factor. The greatest one is 12, giving 2 students and 3 books per group. 6 groups would work but is not the greatest; 60 and 72 are not factors of 24.
  26. 26. 2
    2 is prime (its only factors are 1 and 2) and it is even, so the statement is false. 3 is odd and prime, which fits the claim. 9 and 15 are odd but composite, so they do not test it.

Related practice

Multiplication and division facts: arrays and fact familiesPlace value to 100,000Parentheses and the four operationsDivision with remainders: estimate and check