Factors, Multiples and Primes
Primary 5 · Mathematics
Factors, Multiples and Primes
Lesson 2 of 3
14 min
Mutesi's classroom has 24 desks. Her teacher asks her to line them up in equal rows, with no lonely desk left over at the side.
She tries 4 rows of 6. It works. She tries 3 rows of 8. That works too. She tries 5 rows — and 4 desks are left standing. Five simply refuses to fit into 24.
The numbers that do fit exactly into 24 are called its factors. Today you will find every factor of a number, meet the stubborn numbers that hardly have any factors at all, and learn to break any number down into its building blocks.
A factor of a number divides into it exactly, leaving nothing over.
Guessing factors one by one is risky — you will miss some. The safe way is to hunt in pairs, starting at 1 and walking upwards. Watch with 24.
| Try | Does it fit into 24? | The pair you found |
|---|---|---|
| 1 | Yes — 24 ÷ 1 = 24 | 1 and 24 |
| 2 | Yes — 24 ÷ 2 = 12 | 2 and 12 |
| 3 | Yes — 24 ÷ 3 = 8 | 3 and 8 |
| 4 | Yes — 24 ÷ 4 = 6 | 4 and 6 |
| 5 | No — 4 are left over | none |
| 6 | Yes — but 6 already appeared. Stop! | none new |
Once a number you try is already sitting in an earlier pair, you have caught them all. Now write them in order:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 — eight factors.
Two things are always true: 1 is a factor of every number, and every number is a factor of itself.
Now run the same hunt on 7. Is 7 ÷ 2 exact? No. 7 ÷ 3? No. 7 ÷ 4? 7 ÷ 5? 7 ÷ 6? No, no, no. Only 1 and 7 work.
Seven has exactly two factors. It cannot be built from smaller pieces.
A number with exactly two factors — 1 and itself — is prime.
A number with more than two factors, like 24, is composite. (Composite means "put together from parts".)
These are the primes you will meet again and again. Learn them the way you learned your times tables:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Two surprises hide in that list:
Beware the impostors — numbers that look prime but are not. Use your divisibility rules on them first!
51 = 3 × 17 57 = 3 × 19 91 = 7 × 13
(For 51 and 57, adding the digits gives 6 and 12 — both multiples of 3. The rule caught them instantly.)
Every composite number can be broken open, again and again, until only primes remain. Watch it happen with 90.
90 is even, so a 2 comes out: 90 = 2 × 45
Is 45 prime? Its digits add to 9, so 3 fits: 45 = 3 × 15
Is 15 prime? No: 15 = 3 × 5
Now 3 and 5 are both prime. There is nothing left to break. Gather up every piece you split off:
90 = 2 × 3 × 3 × 5
Always multiply back to check: 2 × 3 = 6, then 6 × 3 = 18, then 18 × 5 = 90. ✓
That is the prime factorisation of 90 — the recipe of primes that builds it. Here is the beautiful part: every number has only one such recipe (apart from the order you write it in). That is why primes are called the building blocks of all numbers.
The trick is to always divide by the smallest prime that fits, and only move up when it stops fitting.
Step 1 Does 2 fit into 84? It is even, so yes. 84 = 2 × 42
Step 2 Does 2 fit into 42? It is even, so yes. 42 = 2 × 21
Step 3 Does 2 fit into 21? No, 21 is odd. Move up to 3. The digits of 21 add to 3, so yes. 21 = 3 × 7
Step 4 3 and 7 are both prime. Stop.
Collect the pieces you pulled off along the way — 2, then 2, then 3, then 7:
84 = 2 × 2 × 3 × 7
Check: 2 × 2 = 4, then 4 × 3 = 12, then 12 × 7 = 84. ✓
When the same prime appears more than once, you may write it shorter using a power: 84 = 2² × 3 × 7. Both ways are correct.
Tip
Never guess — climb the primes. Try 2 first, then 3, then 5, then 7, then 11. Only move up when the smaller prime refuses to fit. Do that and you can never miss a factor and never go round in circles.
Factor or multiple? Factors of 12 are never bigger than 12 (they fit inside it). Multiples of 12 are never smaller than 12 (12 fits inside them). Small word, big difference.
Work each one on paper first, then check.
List all the factors of 36.
Answer: hunt in pairs — 1 and 36, 2 and 18, 3 and 12, 4 and 9, then 6 and 6. Six pairs with itself, so you write it once. 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors.
Is 39 prime or composite?
Answer: add the digits — 3 + 9 = 12, a multiple of 3. So 3 fits: 39 = 3 × 13. It has four factors (1, 3, 13, 39), so it is composite.
Write the prime factorisation of 72.
Answer: 72 = 2 × 36, then 36 = 2 × 18, then 18 = 2 × 9, then 9 = 3 × 3. So 72 = 2 × 2 × 2 × 3 × 3. Check: 2 × 2 × 2 = 8, and 3 × 3 = 9, and 8 × 9 = 72. ✓
Which numbers between 20 and 30 are prime?
Answer: 23 and 29. The rest all fall: 22, 24, 26 and 28 are even; 21 = 3 × 7; 25 = 5 × 5; 27 = 3 × 9.
Ishimwe says: "45 = 5 × 9, so the prime factors of 45 are 5 and 9." Is he right?
Answer: not yet. 5 is prime, but 9 is not — it breaks into 3 × 3. He stopped too early. Keep going: 45 = 3 × 3 × 5. You have only finished when every single piece is prime.
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