Prime Factorization

Split a number into its prime factors with a factor tree, write the answer with powers, and work with numbers already given as primes.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 51 real past-paper questions below

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What Prime Factors Are

A prime number has exactly two factors: 1 and itself. The first few are 2, 3, 5, 7, 11 and 13.

Every whole number above 1 is made by multiplying primes together. Finding those primes is prime factorization.

Worked example

1
$12 = 2 \times 6$ — Split 12 into two factors. 2 is prime, but 6 is not, so keep going.
2
$= 2 \times 2 \times 3$ — Split the 6: 6 = 2 × 3. Now every number is prime, so stop.
3
$= 2^{2} \times 3$ — Two 2s multiplied is 2². Powers make the answer shorter.

Rule

"Product of powers of its prime factors" means: write it with powers, like $2^{2} \times 3$ . This is called index form.

Write a Number as a Product of Primes

The 3-mark question: write a number like 504 as a product of its prime factors. Keep dividing by the smallest prime that goes in.

Worked example

1
$504 \div 2 = 252$ — 504 is even, so the smallest prime, 2, goes in.
2
$252 \div 2 = 126$ — 252 is still even, so divide by 2 again.
3
$126 \div 2 = 63$ — 126 is still even. That is the third 2.
4
$63 \div 3 = 21$ — 63 is odd, so 2 no longer goes in. Try the next prime, 3. Its digits add to 6 + 3 = 9, which is in the 3 times table, so 3 goes in.
5
$21 \div 3 = 7$ — 21 is in the 3 times table, so divide by 3 again.
6
$7 \div 7 = 1$ — 7 is prime, so only 7 goes in. Reaching 1 means you are done.
7
$504 = 2 \times 2 \times 2 \times 3 \times 3 \times 7$ — Write every prime you divided by. This is the product of prime factors.
8
$504 = 2^{3} \times 3^{2} \times 7$ — Three 2s become 2³ and two 3s become 3². This is index form.

Common mistake

Stopping at a number that is not prime, like $2^{3} \times 9 \times 7$ . 9 is $3 \times 3$ , so keep splitting.

Key idea

Check it: multiply your primes back. $8 \times 9 \times 7 = 504$ , so it is right.

Key idea

Write every division down. Most of the marks are for those lines, so a slip at the end still keeps marks. When the question says “show your working clearly”, an answer with no working can score 0.

Big Numbers: Split Off the Hundred

The 3-mark question can use a big number, even one in standard form like $5.4 \times 1 0^{3}$ . Write it out, then split off the tens first.

Worked example

1
$5.4 \times 1 0^{3} = 5400$ — Times $1 0^{3}$ means times 1000, so the digits move 3 places left.
2
$5400 = 54 \times 100$ — It ends in two zeros, so split off 100. Both parts are easy to break up.
3
$54 = 2 \times 27$ — 54 is even, so take out a 2.
4
$27 = 3 \times 3 \times 3 = 3^{3}$ — 27 is three 3s multiplied, so write it as $3^{3}$ .
5
$100 = 10 \times 10 = (2 \times 5) \times (2 \times 5)$ — Each 10 is $2 \times 5$ , because 10 is not prime.
6
$100 = 2^{2} \times 5^{2}$ — Two 2s and two 5s, written with powers.
7
$5400 = 2 \times 3^{3} \times 2^{2} \times 5^{2}$ — Put the pieces from 54 and 100 back together.
8
$5400 = 2^{3} \times 3^{3} \times 5^{2}$ — Join the 2s: $2 \times 2^{2} = 2^{1 + 2} = 2^{3}$ . Each prime must appear only once.

Common mistake

Leaving $1 0^{2}$ in the answer. 10 is not prime; it is $2 \times 5$ .

Key idea

A big power of ten splits the same way: $1 0^{n} = 2^{n} \times 5^{n}$ . For example, $1 0^{6} = 2^{6} \times 5^{6}$ .

Numbers Already Given as Primes

The 2-mark question gives numbers as primes already, like $A$ and $B$ below. Do not work them out. Multiply by adding the powers of each prime.

A = 2^{3} \times 3 \times 5^{2}

B = 2^{2} \times 3^{4}

Worked example

1
$A B = 2^{3} \times 3 \times 5^{2} \times 2^{2} \times 3^{4}$ — Multiplying means writing both lists of primes side by side.
2
$A B = 2^{3 + 2} \times 3^{1 + 4} \times 5^{2}$ — Group the same primes and add their powers. 3 on its own is 3¹, so its power is 1.
3
$A B = 2^{5} \times 3^{5} \times 5^{2}$ — 3 + 2 = 5 and 1 + 4 = 5. Only A has 5s, so $5^{2}$ stays.
4
$12 = 2^{2} \times 3$ — For $12 A$ : 12 is not prime, so split it into primes first.
5
$12 A = 2^{2} \times 3 \times 2^{3} \times 3 \times 5^{2}$ — Write the primes of 12 next to the primes of A.
6
$12 A = 2^{5} \times 3^{2} \times 5^{2}$ — Add the powers: 2 + 3 = 5 and 1 + 1 = 2.

Common mistake

Multiplying the powers: $2^{3} \times 2^{2}$ is $2^{5}$ , not $2^{6}$ .

Key idea

Letters work the same. If $A = 2^{n} \times 5$ , then $4 A = 2^{n + 2} \times 5$ , because 4 is $2^{2}$ .

Square It or Divide It

Exams also square a number given in primes, or divide it. Do not work it out: change the powers.

k = 2^{3} \times 3 \times 7^{2}

Worked example

1
$k^{2} = k \times k$ — Squaring means multiplying k by itself.
2
$k^{2} = 2^{3 + 3} \times 3^{1 + 1} \times 7^{2 + 2}$ — Multiplying adds the powers of each prime, the same as the last card.
3
$k^{2} = 2^{6} \times 3^{2} \times 7^{4}$ — Every power has doubled. So squaring doubles each power.

Now divide $x = 2^{4} \times 3^{3} \times 5 \times 7^{2}$ by 84.

Worked example

1
$84 = 2^{2} \times 3 \times 7$ — Split the number you divide by into primes first.
2
$x \div 84 = 2^{4 - 2} \times 3^{3 - 1} \times 5 \times 7^{2 - 1}$ — Dividing takes away the powers of each prime. 84 has no 5, so the 5 stays.
3
$x \div 84 = 2^{2} \times 3^{2} \times 5 \times 7$ — 4 − 2 = 2, 3 − 1 = 2 and 2 − 1 = 1. A power of 1 is not written.

Common mistake

Squaring the powers: $(2^{3})^{2}$ is $2^{6}$ , not $2^{9}$ . Double the power, do not square it.

Key idea

A prime written as a letter works the same. If $k = 2 \times p^{3}$ , then $k^{2} = 2^{2} \times p^{6}$ .

Which Method? Read the Question

Find the key words, then pick the method.

You see	You do
Product of prime factors	divide by primes, write every prime
Product of powers of its prime factors	same, then use powers
A big number or standard form	write it out, split off the tens
Numbers already given as primes	add the powers of each prime
Find the power, like $72 = 2^{m} \times 3^{2}$	split 72 into primes, read off the power of 2
Square or divide a number given as primes	double every power, or take away powers
Highest common factor or lowest common multiple	HCF and LCM
Smallest number to make a square or cube	Square Numbers
Simplify a square root	Surds

Key idea

Before you finish, check every number in your answer is prime, and each prime appears only once, with a power.

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