Prime Numbers

Know what a prime number is, spot primes in a list, prove a number is not prime, and find primes that fit a rule.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 49 real past-paper questions below

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What a Prime Number Is

A prime number has exactly two factors: 1 and itself. Nothing else divides into it.

Number	Factors
7	1, 7: prime
9	1, 3, 9: not prime
2	1, 2: prime
1	just 1: not prime

Memory trick

The primes up to 30: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ .

Rule

2 is the only even prime. Every other even number divides by 2. And 1 is never prime: it has only one factor.

Key idea

About 2,300 years ago, the Greek mathematician Euclid proved that the primes never run out. His idea: multiply any list of primes and add 1, like $2 \times 3 \times 5 + 1 = 31$ . No prime on your list divides the new number, so there is always a prime you missed.

Spot the Prime in a List

The 1-mark question: pick the prime from a list, or write down the prime between two numbers. Test each number by dividing by 2, 3, 5 and 7.

From 21, 27, 39, 41, 45, 49, write down a prime number.

Worked example

1
$21 = 3 \times 7$ — 3 goes into 21, so 21 has a factor other than 1 and itself. Not prime.
2
$27 = 3 \times 9$ — 2 + 7 = 9 and 3 goes into 9, so 3 goes into 27. Not prime.
3
$39 = 3 \times 13$ — 3 + 9 = 12 and 3 goes into 12, so 3 goes into 39. Not prime.
4
$45 = 5 \times 9$ — It ends in 5, so 5 goes into it. Not prime.
5
$49 = 7 \times 7$ — It is a square number, 7 times itself, so 7 is a factor. Not prime.
6
$41$ — Odd, digits add to 5, does not end in 0 or 5, and 41 ÷ 7 leaves 6. Nothing goes in, so 41 is prime.

Key idea

A quick test for 3: add the digits. For 39, $3 + 9 = 12$ , and 3 goes into 12, so 3 goes into 39.

Common mistake

Picking an odd number because it looks prime. 49 and 91 are odd, but $91 = 7 \times 13$ . Between 90 and 100, only 97 is prime.

Is It Prime? Give a Reason

The 1 or 2-mark question: someone says a number is prime, or that a rule always gives primes. Are they right? To say no, show one number that breaks it.

Sam says 93 is a prime number. Is Sam correct?

Worked example

1
$9 + 3 = 12$ — Add the digits to test for 3. 3 goes into 12, so 3 goes into 93.
2
$93 \div 3 = 31$ — Divide to find the other factor. It goes exactly, nothing left over.
3
$93 = 3 \times 31$ — So 3 is a factor of 93, as well as 1 and 93.
4
$No: 93 = 3 \times 31$ — Write “No” and the reason. The mark is for the factor, not the word.

Lena says $n^{2} + n + 3$ is prime for any whole number $n$ . Try small numbers until one fails.

Worked example

1
$n = 1 : 1 + 1 + 3 = 5$ — Start with a small whole number. 5 is prime, so the rule works so far.
2
$n = 2 : 4 + 2 + 3 = 9$ — Try the next one. $2^{2} = 4$ , then add 2 and 3.
3
$9 = 3 \times 3$ — 9 has a factor other than 1 and 9, so it is not prime. One value has broken the rule.
4
$No: n = 2 gives 9 = 3 \times 3$ — Write the value of n, what the expression gives, and why that is not prime.

Key idea

One example is enough to prove a claim wrong. Someone says no even number is prime? 2 is even and prime, so they are wrong.

Common mistake

Writing just “No”. Show the working: the factor, like $93 = 3 \times 31$ , or the value of $n$ and what it gives. That working is where the marks are.

Bigger Numbers: Test Up to √n

Above 120, testing 2, 3, 5 and 7 is not enough. This rule tells you which primes to try, and when to stop.

Formula

p \times p \leq n

$n$ is the number you are testing. $p$ is each prime you try: 2, 3, 5, 7, 11, 13 … Try every prime whose $p \times p$ is not bigger than $n$ . If none of them divides $n$ , then $n$ is prime.

Find the prime number between 140 and 150.

Worked example

1
$11 \times 11 = 121, 13 \times 13 = 169$ — Every number here is below 169, so the biggest prime to try is 11. Test with 2, 3, 5, 7 and 11.
2
$142, 144, 146, 148$ — These are even, so 2 goes into them. Cross them out.
3
$145 = 5 \times 29$ — It ends in 5, so 5 goes into it. Cross it out.
4
$141 = 3 \times 47$ — 1 + 4 + 1 = 6, so 3 goes into it. Cross it out.
5
$147 = 3 \times 49$ — 1 + 4 + 7 = 12, so 3 goes into it. Cross it out.
6
$143 = 11 \times 13$ — 2, 3, 5 and 7 all fail here, but 11 goes in. This is why you must try 11.
7
$149$ — 2, 3, 5, 7 and 11 do not go in, so 149 is prime.

Memory trick

Up to 120, testing 2, 3, 5 and 7 is enough, because the next prime gives $11 \times 11 = 121$ .

Common mistake

Stopping at 7 for a bigger number. 143 passes 2, 3, 5 and 7, but $143 = 11 \times 13$ .

Find the Primes, Then Use Them

The 2-mark question: find the primes that fit a rule, then add or multiply them.

Two prime numbers are between 30 and 40. Work out their sum.

Worked example

1
$31, 32, 33, \dots, 39$ — List every whole number in the gap, so none is missed.
2
$32, 34, 36, 38$ — Even, so 2 goes into them. Cross them out.
3
$33 = 3 \times 11$ — 3 + 3 = 6, so 3 goes into 33. Cross it out.
4
$35 = 5 \times 7$ — It ends in 5, so 5 goes into it. Cross it out.
5
$39 = 3 \times 13$ — 3 + 9 = 12, so 3 goes into 39. Cross it out.
6
$31, 37$ — Only these survive: 2, 3, 5 and 7 do not go in. They are the two primes.
7
$31 + 37 = 68$ — Add the two primes to get the sum.

Find the sum of all the prime numbers less than 20.

Worked example

1
$2, 3, 5, 7$ — The primes below 10. Start with 2: it is the only even prime.
2
$11, 13, 17, 19$ — The primes from 10 to 19. 15 = 3 × 5 is not one.
3
$2 + 3 + 5 + 7 = 17$ — Add the first group.
4
$11 + 13 + 17 + 19 = 60$ — Add the second group.
5
$17 + 60 = 77$ — Add the two totals. The sum is 77.

Key idea

Write the primes down before you add them. If the total slips, the primes you found can still earn a mark.

Common mistake

Missing one or adding an extra, like 1 or 15. Leaving out 2 is the most common slip: 2 is prime. Check every number in the gap.

Primes That Fit Clues

Some questions describe a prime with clues. Read each clue, list what fits, then check every clue.

Rule

$20 < P < 28$ means P is more than 20 and less than 28. The end numbers 20 and 28 are not allowed.

P is a prime number and $20 < P < 28$ . Find P.

Worked example

1
$21, 22, \dots, 27$ — P is between 20 and 28, so list the whole numbers in between.
2
$22, 24, 26$ — Even, so 2 goes into them. Cross them out.
3
$21 = 3 \times 7$ — 3 goes into 21. Cross it out.
4
$25 = 5 \times 5$ — It ends in 5, so 5 goes into it. Cross it out.
5
$27 = 3 \times 9$ — 2 + 7 = 9, so 3 goes into 27. Cross it out.
6
$P = 23$ — Only 23 is left, and 2, 3 and 5 do not go in. So P = 23.

P is a prime number and $40 < P < 70$ . P is 3 less than a square number. Find P.

Worked example

1
$49, 64$ — P is between 40 and 70, so P + 3 is between 43 and 73. The only square numbers there are 7² = 49 and 8² = 64.
2
$49 - 3 = 46$ — Take 3 away to get a possible P. 46 is even, so it is not prime.
3
$64 - 3 = 61$ — Take 3 away again. 2, 3, 5 and 7 do not go into 61, so it is prime.
4
$P = 61$ — Check every clue: 61 is prime, it is between 40 and 70, and 61 + 3 = 64 is a square.

Key idea

Write down the square numbers you used. In a 2-mark question, a correct square seen can earn a mark even if P is wrong.

Common mistake

Using an end number. In $20 < P < 29$ , P cannot be 29, even though 29 is prime. The sign < means “less than”, not “equal to”.

Which Method? Read the Question

The words tell you what to do.

You see	You do
Write down a prime	test each number with 2, 3, 5, 7
The prime between	check each number in the gap
Is this correct? Give a reason	show a factor, or a value that fails
Sum of these primes	find them all, then add
A prime above 120	also try 11, 13, … while p × p ≤ n
A prime with clues, like 20 < P < 28	list what fits, check every clue
Product of prime factors	Prime Factorization
Highest common factor	HCF and LCM
Pick a square, cube or odd	Integers

Key idea

Up to 120, you only need to test 2, 3, 5 and 7. If none go in, the number is prime. Above 120, keep going with 11, 13, … until $p \times p$ is bigger than the number.

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