Factors, Multiples & Primes – Grade 5 Maths

Factors

All pairs that multiply to make a number

Multiples

Skip counting sequences

Prime Numbers

Exactly 2 factors: 1 and itself

Squares & Cubes

1, 4, 9, 16... and 1, 8, 27...

HCF & LCM

Highest Common Factor and Lowest Common Multiple

1. Factors

A factor of a number divides into it exactly (with no remainder). Every number has at least two factors: 1 and itself.

Factor pairs of 24: 1×24, 2×12, 3×8, 4×6 → Factors: 1, 2, 3, 4, 6, 8, 12, 24

To find all factors of 36:
1 × 36 = 36 ✓
2 × 18 = 36 ✓
3 × 12 = 36 ✓
4 × 9 = 36 ✓
6 × 6 = 36 ✓
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors total)

💡 Always check in pairs from 1 upward. Stop when your two numbers meet or cross in the middle.

2. Multiples

Multiples of a number are what you get when you multiply it by 1, 2, 3, 4, 5... They go on forever!

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70...

Multiples of 6: 6, 12, 18, 24, 30, 36...
Multiples of 9: 9, 18, 27, 36, 45, 54...

Notice that 18 and 36 appear in both lists — these are common multiples of 6 and 9.

💡 A number is a multiple of another if the other divides into it exactly. Is 42 a multiple of 7? Yes, because 42 ÷ 7 = 6 with no remainder.

3. Prime Numbers

A prime number has exactly two factors: 1 and itself. The number 1 is NOT prime (it has only one factor).

Primes to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

2357 11131719 23293137 414347

Prime factorisation means writing a number as a product of prime numbers.
36 = 2 × 2 × 3 × 3 = 2² × 3²
Use a factor tree: 36 → 4 × 9 → 2×2 × 3×3

💡 2 is the only even prime number. All other even numbers have 2 as a factor, so they can't be prime.

4. Square and Cube Numbers

A square number is made by multiplying a whole number by itself: n × n = n²

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

1²=12²=43²=94²=16 5²=256²=367²=498²=64 9²=8110²=10011²=12112²=144

A cube number is made by multiplying a whole number by itself three times: n × n × n = n³

Cube numbers: 1, 8, 27, 64, 125, 216

1³=12³=83³=27 4³=645³=1256³=216

💡 64 is both a square number (8²) and a cube number (4³). Can you find another? (Answer: 1!)

5. HCF and LCM

The Highest Common Factor (HCF) is the largest factor that two numbers share.

The Lowest Common Multiple (LCM) is the smallest multiple that two numbers share.

HCF: list factors of each number, find the highest one in common.

HCF of 12 and 18:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6 → HCF = 6

LCM: list multiples of each number, find the first one in common.

LCM of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24...
First common multiple: LCM = 12

💡 HCF is useful for simplifying fractions. LCM is useful for finding common denominators and "when do two things happen at the same time" problems.

Example 1 — Factor Pairs of 36

Start with 1: 1 × 36 = 36 ✓

Try 2: 2 × 18 = 36 ✓. Try 3: 3 × 12 = 36 ✓

Try 4: 4 × 9 = 36 ✓. Try 5: 36 ÷ 5 = 7.2 — not whole, skip.

Try 6: 6 × 6 = 36 ✓. Next would be 7: 36 ÷ 7 is not whole. Stop here (we've crossed the middle).

All factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Example 2 — Prime Factorisation of 60 (Factor Tree)

60 = 6 × 10

6 = 2 × 3 (both prime!)

10 = 2 × 5 (both prime!)

60 = 2 × 3 × 2 × 5 = 2² × 3 × 5

Total number of prime factors (with repeats): 4 (two 2s, one 3, one 5)

Example 3 — HCF and LCM of 12 and 18

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

Common factors: 1, 2, 3, 6 → HCF = 6

Multiples of 12: 12, 24, 36, 48...

Multiples of 18: 18, 36, 54...

LCM = 36

Example 4 — Square & Cube Numbers

7² = 7 × 7 = 49 (square number)

4³ = 4 × 4 × 4 = 16 × 4 = 64 (cube number)

Is 49 prime? No — it has 3 factors: 1, 7, 49

Is 64 a square? Yes (8² = 64). Is it also a cube? Yes (4³ = 64).

🔬 Visualizer 1 — Factor Finder

Enter a number (up to 200) and see all its factors and factor pairs.

Number

🔬 Visualizer 2 — HCF & LCM Calculator

Enter two numbers to find their HCF and LCM.

Number A

Number B

Exercise 1 — Listing Factors

For each question, enter the total number of factors the number has.

1. How many factors does 12 have? (1,2,3,4,6,12)

2. How many factors does 24 have?

3. How many factors does 36 have?

4. What is the largest factor of 48 that is less than 48?

5. What is the largest factor of 60 that is less than 60?

6. How many factor pairs does 16 have? (1×16, 2×8, 4×4)

7. What is the second largest factor of 30?

8. How many factors does 100 have?

9. Which factor of 18 is between 3 and 9?

10. What is the sum of all factors of 15? (1+3+5+15)

Exercise 2 — Multiples and Prime Numbers

1. What is the 7th multiple of 8?

2. What is the 9th multiple of 7?

3. Is 91 prime? Enter 1 for yes, 0 for no. (Try 7 × 13)

4. How many prime numbers are there between 1 and 20?

5. What is the smallest prime number greater than 40?

6. Is 97 prime? Enter 1 for yes, 0 for no.

7. What is the 6th multiple of 9?

8. Which of these is a multiple of both 3 and 5: 25, 45, 35? Enter the number.

9. What is the largest prime number less than 50?

10. How many prime numbers are there between 20 and 40?

Exercise 3 — Square and Cube Numbers

1. 7² = ?

2. 11² = ?

3. 12² = ?

4. 3³ = ?

5. 4³ = ?

6. 5³ = ?

7. What square number comes between 80 and 90?

8. √64 = ? (square root of 64)

9. Is 50 a square number? Enter 1 for yes, 0 for no.

10. What is the sum of the first three cube numbers? (1 + 8 + 27)

Exercise 4 — Prime Factorisation

Write each number as a product of primes using a factor tree. Answer as requested.

1. 12 = 2² × 3. How many prime factors does 12 have in total (with repeats)? (2,2,3)

2. 18 = 2 × 3². How many prime factors does 18 have in total (with repeats)?

3. 60 = 2² × 3 × 5. How many prime factors in total?

4. 36 = 2² × 3². What is the largest prime factor of 36?

5. What is the largest prime factor of 84? (84 = 4 × 21 = 2² × 3 × 7)

6. What is the largest prime factor of 45? (45 = 9 × 5 = 3² × 5)

7. 24 = 2³ × 3. How many prime factors in total (with repeats)?

8. 100 = 2² × 5². What is the sum of the prime base numbers? (2 + 5)

9. 48 = 2⁴ × 3. How many prime factors in total?

10. What is the largest prime factor of 90? (90 = 2 × 3² × 5)

Exercise 5 — HCF and LCM

1. HCF of 12 and 18 = ?

2. HCF of 24 and 36 = ?

3. HCF of 15 and 25 = ?

4. LCM of 4 and 6 = ?

5. LCM of 5 and 8 = ?

6. LCM of 6 and 9 = ?

7. HCF of 48 and 60 = ?

8. LCM of 8 and 12 = ?

9. HCF of 30 and 45 = ?

10. LCM of 7 and 11 = ?

🏋️ Practice — 20 Questions

1. How many factors does 20 have?

2. 8² = ?

3. 2³ = ?

4. What is the 5th multiple of 11?

5. HCF of 16 and 24 = ?

6. LCM of 3 and 7 = ?

7. Is 51 prime? (51 = 3 × 17) Enter 1=yes, 0=no.

8. What is the largest factor of 72 less than 72?

9. 6³ = ?

10. What is the smallest prime number?

11. LCM of 4 and 10 = ?

12. What square number is between 35 and 45?

13. HCF of 9 and 21 = ?

14. What is the 8th multiple of 6?

15. 10² = ?

16. How many factors does 48 have?

17. Is 1 a prime number? Enter 1=yes, 0=no.

18. LCM of 6 and 8 = ?

19. What is the largest prime factor of 30?

20. Sum of first 4 square numbers (1+4+9+16) = ?

🏆 Challenge — 8 Questions

1. A bell rings every 4 minutes and a whistle blows every 6 minutes. They both sound together at 12:00. After how many minutes will they next both sound together? (LCM)

2. What is the smallest number that has exactly 3 factors? (Hint: it's a square of a prime)

3. Express 36 as a sum of two prime numbers. Enter the larger prime.

4. Two buses leave a stop together. Bus A comes every 8 minutes, Bus B every 12 minutes. After how many minutes do they leave together again?

5. What is the smallest number with exactly 6 factors? (Try 12: factors 1,2,3,4,6,12)

6. What is the HCF of 84 and 126?

7. What is the LCM of 12, 15, and 20?

8. Express 50 as a sum of two prime numbers. Enter the larger prime.

🔢 Factors, Multiples & Primes