Factors, Multiples, Primes, Squares and Cubes

This topic is about five kinds of whole number: factors, multiples, prime numbers, square numbers and cube numbers. Each one is explained below, with a worked example.

In the exam you are often asked to write down one number of the right kind, for 1 mark. It might be a factor of a number, a multiple between two values, or the square number or prime number in a list.

Questions worth 2 or 3 marks ask for all the prime numbers in a range, the highest common factor (HCF) or lowest common multiple (LCM) of two numbers, or a short problem about packs of things or events that repeat.

A factor of a number divides into it exactly, with no remainder. 3 is a factor of 24, because 24 \div 3 = 8. 5 is not a factor of 24, because 24 \div 5 leaves a remainder.

Factors come in factor pairs: two numbers that multiply to make the number. 4 \times 6 = 24, so 4 and 6 are a factor pair of 24. You can picture a factor pair as squares in an exact rectangle.

To find all the factors, work in pairs. Start with 1 times the number, then try 2, 3, 4, and so on. Stop when the pairs start to repeat. 1 and the number itself are always factors.

A multiple of a number is in its times table. The multiples of 6 are 6,\ 12,\ 18,\ 24,\ 30 and so on, for ever.

Factors and multiples are easy to mix up. A factor divides into the number. A multiple is a number that the number divides into. 4 is a factor of 12, and 12 is a multiple of 4.

When a question says “between” two numbers, check your answer is bigger than the first number and smaller than the second.

A prime number has exactly two factors: 1 and itself. 13 is prime, because its only factors are 1 and 13. 15 is not prime, because 3 \times 5 = 15.

The prime numbers up to 30 are 2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29

1 is not a prime number, because it has only one factor. 2 is the only even prime number: every other even number has 2 as a factor as well.

Not every odd number is prime. To test an odd number up to 100, try dividing it by 3, 5 and 7. For example, 21 = 3 \times 7 and 27 = 3 \times 9, so neither of them is prime.

A square number is a whole number multiplied by itself. 4 \times 4 = 16, so 16 is a square number. We write 4^2 = 16 and say “four squared”. The small 2 means the number is multiplied by itself, not by 2.

Learn the square numbers up to 15^2: 1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ 121,\ 144,\ 169,\ 196,\ 225. 1 is a square number too, because 1 \times 1 = 1.

A square root undoes squaring. \sqrt{81} = 9, because 9 \times 9 = 81. It is not 81 \div 2.

A cube number is a whole number multiplied by itself, and then by itself again. 2^3 = 2 \times 2 \times 2 = 8, so 8 is a cube number. The small 3 means three of the number multiplied together, not \times 3.

Learn these cube numbers: 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125 and 10^3 = 1000

A cube root undoes cubing. \sqrt[3]{8} = 2, because 2 \times 2 \times 2 = 8. Do not divide by 3.

A common factor is a factor of two numbers at once. The highest common factor (HCF) is the biggest one. To find it, list the factors of each number, then pick the biggest number that is in both lists.

A common multiple is in the times tables of two numbers at once. The lowest common multiple (LCM) is the smallest one. To find it, list the multiples of each number until the same number appears in both lists.

The two are easy to mix up. The HCF divides into both numbers, so it is never bigger than the smaller number. The LCM is a multiple of both numbers, so it is never smaller than the bigger number.

Use the HCF when you share things into the greatest number of equal groups with nothing left over. Use the LCM for things that repeat, such as buses or lights, to find when they happen together again. The LCM also helps when two items come in different pack sizes and you want the same number of each.

• Mixing up factors and multiples. A factor divides into the number; a multiple is in its times table.
• Finding the LCM when the HCF is asked for. The HCF divides into both numbers, so it is never bigger than the smaller number.
• Thinking every odd number is prime. 21, 27, 33 and 39 are odd but not prime, so test by dividing by 3, 5 and 7.
• Calling 1 a prime number. 1 has only one factor, so it is not prime, but it is a square number.
• Doubling to square, halving for a square root, or dividing by 3 for a cube root. 6^2 = 36, not 12; \sqrt{36} = 6, not 18; \sqrt[3]{64} = 4, not 64 \div 3.
• Giving a number outside the range. Check the answer is between the two numbers given and is the right kind of number.