GCSE Maths · Factors, Multiples and Primes · Higher · Summary

GCSE Maths Factors, Multiples and Primes (Higher): summary and key words

Part of the GCSE Maths factors, multiples and primes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Multiples and factors	Multiples: the times table of the number. Factors: find pairs that multiply to the number. Use divisibility tests (2, 3, 4, 5, 6, 8, 9, 10).
Prime numbers	Exactly two factors, 1 and itself. 1 is not prime; 2 is the only even prime. To test: divide by primes up to the square root.
Prime factorisation	Factor tree or division ladder, then index form: 360 = 2³ × 3² × 5. Every number has only one factorisation.
HCF and LCM	List, or write as primes and use a Venn diagram: HCF = multiply the middle; LCM = multiply everything. HCF × LCM = product of the two numbers.
Systematic listing	Fix the first choice, list the rest in order, then count. Numbers in order of size.
Product rule (Higher)	Multiply the number of choices at each stage. Fill any restricted position first.
Squares, cubes, roots	Learn squares to 15², cubes 1, 8, 27, 64, 125, 1000, and powers of 2, 3, 4, 5. √ and the cube root undo the powers.
Laws of indices	Same base: add the indices to multiply, subtract to divide, multiply for a power of a power.
Zero and negative indices	a⁰ = 1 and a⁻ⁿ = 1 ÷ aⁿ. (Higher) (a/b)⁻ⁿ = (b/a)ⁿ.
Estimating roots and powers	Find the squares (cubes) either side. For 1 d.p., test one-decimal-place numbers and the halfway value. Round the base to estimate a power.

Can you…?

I can list the multiples and factors of a number and use divisibility tests.
I can say whether a number is prime and list the primes up to 50.
I can write a number as a product of prime factors in index form.
I can find the HCF and LCM by listing and with a Venn diagram, and use them to solve problems.
I can list outcomes systematically and count them.
(Higher) I can use the product rule for counting.
I know the squares to 15², the common cubes and the powers of 2, 3, 4 and 5, and I can find roots.
I can use the laws of indices and the rules for zero and negative indices.
I can estimate square and cube roots and use a calculator for powers and roots.
(Higher) I can find the number of factors, make a number a perfect square, and work with fractional bases and negative indices.

Key terms

Multiple: A number in the times table of another number, e.g. 12 is a multiple of 3.
Factor: A whole number that divides exactly into another, e.g. 4 is a factor of 12.
Common factor / common multiple: A factor (or multiple) shared by two or more numbers.
Prime number: A number with exactly two factors, 1 and itself.
Composite number: A number with more than two factors.
Prime factorisation: Writing a number as a product of prime numbers only.
Index form: Writing repeated multiplication as a power, e.g. 2 × 2 × 2 = 2³.
Unique factorisation theorem: Every whole number above 1 has exactly one prime factorisation, apart from the order.
HCF: Highest common factor: the biggest number that divides into two or more numbers.
LCM: Lowest common multiple: the smallest number in all of the times tables.
Venn diagram: A diagram of overlapping circles; here it shows the prime factors that two numbers share.
Systematic listing: Listing possibilities in a planned order so that none is missed or repeated.
Product rule for counting: If there are m ways of doing one task and n ways of doing another, there are m × n ways of doing both.
Square number: The result of multiplying a whole number by itself: 1, 4, 9, 16 ...
Cube number: The result of multiplying a whole number by itself twice more: 1, 8, 27, 64 ...
Square root / cube root: The number that, squared (cubed), gives the original number.
Index (power, exponent): The small number that says how many copies of the base are multiplied.
Base: The number that is being multiplied by itself in a power.
Reciprocal: 1 divided by a number. The reciprocal of 4 is 14.
Trial and improvement: Trying values, checking whether they are too big or too small, and adjusting.