Prime Numbers
Recognising primes, listing primes, and applied prime reasoning · Grade 1–4.
This booklet helps you learn about prime numbersA whole number bigger than 1 with exactly 2 factors: 1 and itself., step by step.
What’s a prime number?
A prime numberA whole number bigger than 1 with exactly 2 factors: 1 and itself. is a whole numberA number with no fractional part: 0, 1, 2, 3, 4, … bigger than 1 that has ONLY two factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.: 1 and itself.
Example: 7 is primeA whole number bigger than 1 with exactly 2 factors: 1 and itself., because the only numbers that fit exactly into 7 are 1 and 7. Nothing else divides 7 without leaving a remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1..
The first few primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. are: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
Note: 1 is NOT a prime numberA whole number bigger than 1 with exactly 2 factors: 1 and itself. — it only has one factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. (itself), not two.
What you’ll learn
We start by recognising small primesA whole number bigger than 1 with exactly 2 factors: 1 and itself.. Then we tell primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. apart from composite numbersA whole number with more than 2 factors. Opposite of prime. (1 is neither prime nor composite.) (numbers that have more than two factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.). Every whole numberA number with no fractional part: 0, 1, 2, 3, 4, … bigger than 1 is either primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. or compositeA whole number with more than 2 factors. Opposite of prime. (1 is neither prime nor composite.).
How to use the booklet
Work through the questions in order.
If you get stuck, that’s OK — stop there. Where you stop tells your tutor exactly where you need help.
What a prime number is
Aim: Recognise primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. by counting their factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
A whole number is prime when it has exactly two factors — 1 and itself — and nothing else divides into it. That one rule is the whole idea, and every question in this paper comes back to it.
Two facts follow straight from the rule:
To test a number by hand, list its factors and count them. The next two examples show the two possible outcomes.
- 1 and 7 are factors, as they always are for any number.
- Now try everything in between — 2, 3, 4, 5, 6. None divides 7 exactlyGoes in exactly, with no remainder. 36 is divisible by 4 because 36 ÷ 4 = 9. (7 ÷ 2 = 3.5, 7 ÷ 3 ≈ 2.33, and so on).
- So the only factors are 1 and 7 — exactly two. 7 is prime.
- 1 and 6 are factors, as always.
- Now try the numbers in between: 6 ÷ 2 = 3 and 6 ÷ 3 = 2, both whole — so 2 and 3 are factors too.
- That makes 1, 2, 3, 6 — four factors, not two. 6 is not prime. A number with more than two factors is called compositeA whole number with more than 2 factors. Opposite of prime. (1 is neither prime nor composite.).
Try one
Is 11 prime?
Show answer
The first ten primes — know these by heart
Aim: Recall the first ten primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. without working them out.
These turn up so often that it is worth knowing them on sight, instead of testing each one every time. They are easiest to remember in three small groups:
2, 3, 5, 7 · 11, 13, 17, 19 · 23, 29
— the single digits, then the teens, then the twenties.
One pattern also helps you spot larger primes: apart from 2 and 5, every prime ends in 1, 3, 7 or 9. That makes sense, because anything ending in 0, 2, 4, 6 or 8 is evenA whole number divisible by 2. The even numbers are 2, 4, 6, 8, … (so divides by 2), and anything ending in 5 divides by 5.
A prime above 5 must end in 1, 3, 7 or 9 — that part is true. But the reverse is not: ending in one of those digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. does not make a number prime.
9 = 3 × 3, 21 = 3 × 7, 27 = 3 × 9, 33 = 3 × 11, 49 = 7 × 7 — each ends in 1, 3, 7 or 9, yet none is prime.
So treat the last digit as a clue, not a guarantee — when in doubt, test the number (Lesson 3).
Try one
Which of these are prime: 11, 15, 21, 23?
Show answer
Testing whether a number is prime
Aim: Decide whether any number under 100 is primeA whole number bigger than 1 with exactly 2 factors: 1 and itself..
In Lesson 2 you learned the first ten primes by heart, which covers every number up to 29. Beyond that, memory runs out: is 73 prime? Is 91? To settle a bigger number for certain, you need a method that always works.
The idea. A prime has just two factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. — 1 and itself. So testing a number is really a hunt for a third factor. If you can find one, the number is not prime; if there genuinely isn't one, it is.
The method. Divide the number by each prime in turn — 2, then 3, then 5, then 7, and so on. The moment one divides exactlyGoes in exactly, with no remainder. 36 is divisible by 4 because 36 ÷ 4 = 9., you have found a factor, so the number is not prime and you can stop. If you work through the primes and none divide it, the number is prime.
• Even number? Then it divides by 2.
• Ends in 0 or 5? Then it divides by 5.
• DigitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. add up to a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3? Then it divides by 3. (For 87: 8 + 7 = 15, and 15 is a multiple of 3, so 87 divides by 3.)
- 49 is odd, so 2 is out. It doesn't end in 0 or 5, so 5 is out. Its digits 4 + 9 = 13, not a multiple of 3, so 3 is out.
- Next prime is 7: 49 ÷ 7 = 7 exactly — a factor!
- So 49 = 7 × 7, which means 49 is not prime.
But when do you stop? For 49 you got lucky and found a factor quickly. For a number like 53, none of 2, 3, 5 or 7 divides it — so do you keep going to 11, 13, 17, for ever? No. There is a natural place to stop.
What is √n? It is the number that multiplies by itself to give n — for example √49 = 7 because 7 × 7 = 49. You don't need it exactly; a rough value is fine (√53 is about 7.3, so check primes up to 7).
Why it works. Look at the factor pairsTwo numbers that multiply to give a product. 1 × 36 and 4 × 9 are both factor pairs of 36. of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. The pairs meet in the middle at 6, and 6 = √36. After that, the same pairs simply repeat in reverse. So every factor a number has is partnered with one no smaller than √n — which means if nothing up to √n divides it, nothing larger will either.
- 53 is odd, so 2 is out. It doesn't end in 0 or 5, so 5 is out. Its digits 5 + 3 = 8, not a multiple of 3, so 3 is out.
- Next prime is 7: 53 ÷ 7 ≈ 7.57 — not a whole number, so 7 is out too.
- √53 is about 7.3, so 7 was the last prime worth checking. Nothing divided 53, so 53 is prime.
Try one
Is 77 prime?
Show answer
Giving a reason
Aim: Give a reason why a number is not primeA whole number bigger than 1 with exactly 2 factors: 1 and itself., in one short sentence.
Exam questions often ask: “Give a reason why X is not a prime number.” You already did the hard part in Lesson 3 — finding a factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.. Here you simply state that factor as your reason.
Good answer: 27 = 3 × 9, so 3 is a factor other than 1 and 27. That is all the examiner needs.
These answers earn nothing:
- “Because it’s oddA whole number not divisible by 2. The odd numbers are 1, 3, 5, 7, 9, ….” — 7, 11 and 13 are odd and prime, so being odd proves nothing.
- “Because it doesn’t end in an even digit.” — nor do 11, 13 and 17, and they are prime.
- “Because it’s big.” — 29 is bigger still and is prime, so size is irrelevant.
Each weak answer only describes the number; only a factor proves it is not prime.
Try one
Give a reason why 51 is not prime.
Show answer
The sieve of Eratosthenes
Aim: Find all primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. in a range using a visual method.
Eratosthenes (say it: eh-rah-TOSS-thuh-neez) was an ancient Greek mathematician who found a neat way to list every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. up to a chosen number. It works like a sieveA method for finding all primes up to a number by crossing out multiples one by one.: you keep the primes and “sift out” everything else, crossing off multiples one prime at a time.
How far do you sieve? This is the same idea as the √n rule in Lesson 3. To catch every prime up to 100, you only need to sieve with the primes up to √100 = 10 — that is just 2, 3, 5 and 7. Once their multiples are gone, every number still standing must be prime.
The steps, for 1 to 100:
- Write the numbers 1 to 100 in a grid, then cross out 1 — it isn’t prime, as it has only one factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
- The first number left is 2. Ring it, then cross out every multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 2 after it: 4, 6, 8, 10, … Every even number except 2 is now gone.
- The next number not crossed out is 3. Ring it, then cross out its multiples. The new ones are the odd multiples — 9, 15, 21, 27, … (6, 12 and 18 were already crossed by 2).
- The next survivor is 5. Ring it and cross out its multiples. Most, like 10, 15 and 20, are already gone, so only a few are new: 25, 35, 55, …
- The next survivor is 7. Ring it and cross out its multiples. Almost all are already crossed — the only new ones below 100 are 49, 77 and 91, the very numbers that catch people out in Lesson 3.
- Stop here. You have now sieved with every prime up to √100, so every number still uncrossed is prime.
In the finished grid below the primes are gold and bold, and every compositeA whole number with more than 2 factors. Opposite of prime. (1 is neither prime nor composite.) number is crossed out — exactly how you would mark them by hand. The 25 primes between 1 and 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
- Prime
- A whole number with exactly 2 factors: 1 and itself.
- Composite
- A whole number with more than 2 factors. (1 is neither prime nor composite.)
- Factor
- A number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4 exactly.
- Multiple
- A number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. The multiples of 3 are 3, 6, 9, 12, 15, …
- Divisible by
- "36 is divisible by 4" means 36 ÷ 4 gives a whole number (no remainder).
Recap before you start the paper
- PrimeA whole number bigger than 1 with exactly 2 factors: 1 and itself. = exactly 2 factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. (1 and itself).
- 1 is NOT primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. (only 1 factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.).
- 2 IS the only evenA whole number divisible by 2. The even numbers are 2, 4, 6, 8, … primeA whole number bigger than 1 with exactly 2 factors: 1 and itself..
- First ten primesA whole number bigger than 1 with exactly 2 factors: 1 and itself.: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
- To test n: divide by primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. up to √n. If none divide, n is primeA whole number bigger than 1 with exactly 2 factors: 1 and itself..
- Common traps: 9, 21, 25, 27 (look primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. but compositeA whole number with more than 2 factors. Opposite of prime. (1 is neither prime nor composite.)), 49 and 91 (need to check up to 7).
- To justify "not primeA whole number bigger than 1 with exactly 2 factors: 1 and itself.": state a factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.. “X = a × b”.
Now try the practice questions
45 questions across four sections. No time pressure. The step-by-step walkthroughs unlock after you mark.
Instructions
- Type each numeric answer in the box. For sign questions, use a minus sign (e.g.
-7). For radio questions, tap your choice. - Work through the questions carefully. There is no time pressure.
- Your tutor is looking for understanding, method, and the point where you begin to struggle.
- When finished, click Mark My Work. Your answers will then be locked, the correct answers will appear under each wrong question with the working, and you will get 5 similar practice questions inline for each one you missed.
No time pressure
Work carefully. Your tutor is looking for understanding, method, and the point where you begin to struggle.
Recognise primes, recall the first ten, test medium numbers, and identify factors.
Recognise prime numbers and explain what makes a number prime.
- recall the first ten primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)
- decide whether a given number is prime by checking its factors
- explain why 1 is not prime and why 2 is the only even prime
- test small and medium numbers for primality
Word problems and constructing primes from digits.
Apply prime number reasoning to word problems.
- construct primes from a given set of digits
- count or list primes within a given range
- recognise a word problem that needs prime reasoning
- solve problems involving prime properties (e.g. “the largest two-digit prime”)
Stretch: sums and products of primes — links to arithmetic.
Use primes in cross-topic arithmetic problems.
- find sums and products of small primes mentally
- recognise patterns in prime arithmetic
- combine prime knowledge with multi-step calculation
Stretch: word problems combining prime-thinking with arithmetic and logic.
Apply primes to real-life and logic word problems.
- solve real-life problems where primes appear as a constraint
- combine prime reasoning with multi-step problem-solving
- check that an answer is itself prime when the problem demands it