Every whole number has a hidden personality — the way it can (and can't) be split into equal groups. Once you can see those splits, numbers stop being random and start fitting together like puzzle pieces.
Factors: the numbers that divide evenly
A factor of a number is a whole number that divides into it exactly, leaving no remainder. Think of it as a fair share. If you have 12 cookies, you can split them into 3 equal plates of 4 — so 3 and 4 are both factors of 12. But you cannot split 12 cookies into 5 equal plates without breaking a cookie, so 5 is not a factor of 12.
Here is the key test: a whole number \( f \) is a factor of \( n \) when \( n \div f \) comes out even, with nothing left over. For example, \( 12 \div 4 = 3 \) exactly, so 4 is a factor. And \( 12 \div 5 = 2 \) remainder \( 2 \), so 5 is not.
In words A factor goes into a number; a multiple comes out of it. Factors of 12 are 1, 2, 3, 4, 6, 12. Multiples of 12 are 12, 24, 36, 48, … — the numbers you reach by counting up in twelves.
Multiples: counting in steps
A multiple of a number is what you get when you multiply it by 1, 2, 3, and so on. The multiples of 5 are \( 5, 10, 15, 20, 25, \dots \) — the numbers you land on when skip-counting by fives. There is no largest multiple; the list goes on forever. Notice the mirror: 4 is a factor of 12, and 12 is a multiple of 4. Factor and multiple are two views of the same fact, \( 4 \times 3 = 12 \).
Finding all factors — work in pairs
The neat trick for listing every factor is that factors always come in pairs that multiply to give the number. So you hunt for pairs, starting from 1 and climbing, until the two numbers in the pair meet in the middle.
- Start at 1: \( 1 \times 24 = 24 \). So 1 and 24 are factors.
- Try 2: \( 2 \times 12 = 24 \). So 2 and 12 are factors.
- Try 3: \( 3 \times 8 = 24 \). So 3 and 8 are factors.
- Try 4: \( 4 \times 6 = 24 \). So 4 and 6 are factors.
- Try 5: \( 24 \div 5 \) leaves a remainder, so 5 is not a factor — skip it.
- Try 6: but we already found 6 (paired with 4). The pairs have met in the middle, so we can stop.
Reading the pairs in order gives all eight factors: \( 1, 2, 3, 4, 6, 8, 12, 24 \). Working in pairs guarantees you don't miss any and don't double-count.
Primes and composites
Some numbers have lots of factors; others have almost none. A prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first few primes are \( 2, 3, 5, 7, 11, 13, 17, 19, \dots \) Notice 2 is the only even prime — every other even number has 2 as an extra factor.
A composite number is a whole number greater than 1 with more than two factors — it can be built by multiplying smaller numbers. For instance, 24 is composite because, as we just saw, it has eight factors.
What about 1? The number 1 is neither prime nor composite. A prime needs exactly two different factors, but 1 has only one factor — itself. It sits in a category all its own.
Looks can deceive. 51 feels like it might be prime, but \( 51 = 3 \times 17 \), so it has the extra factors 3 and 17 and is actually composite. When checking, it's enough to test small primes (2, 3, 5, 7, …) up to where they'd pair past the square root.
Prime factorization — the atoms of numbers
Here is the big idea. Every whole number greater than 1 can be written as a product of primes, and — apart from the order — there is only one way to do it. This is so important it has a grand name: the Fundamental Theorem of Arithmetic. Primes are the atoms; every other number is a molecule built from them.
A friendly way to find the primes is a factor tree: split the number into any two factors, then keep splitting each branch until every tip is prime.
- Split 60 into a factor pair, say \( 60 = 6 \times 10 \).
- Split each branch: \( 6 = 2 \times 3 \) and \( 10 = 2 \times 5 \).
- Now every tip — 2, 3, 2, 5 — is prime, so the tree is finished.
- Collect the tips: \( 60 = 2 \times 3 \times 2 \times 5 \).
- Group the repeats and write neatly: \( 60 = 2^2 \times 3 \times 5 \).
Try a different start, like \( 60 = 4 \times 15 \), and you'll land on the very same primes — that's the uniqueness promised by the theorem. Check it back: \( 2^2 \times 3 \times 5 = 4 \times 15 = 60 \). ✓
Tip Primes are the building blocks of all numbers. Just as every word is spelled from a fixed alphabet, every whole number is "spelled" from primes — and that spelling is one of a kind. Master prime factorization now and later topics like fractions, GCF, and LCM almost solve themselves.
A first taste of GCF and LCM
Prime factors quietly power two everyday tools. The greatest common factor (GCF) of two numbers is the largest factor they share — handy for simplifying fractions. The lowest common multiple (LCM) is the smallest number that's a multiple of both — handy for adding fractions. For 8 and 12: their shared factors are 1, 2, 4, so the GCF is \( 4 \); their multiples first meet at \( 24 \), so the LCM is \( 24 \). We'll explore these fully later — for now, just notice they both grow straight out of factors and multiples.
Practice
Try each one yourself, then reveal the full solution.
1. List all the factors of 36.
Hunt for factor pairs, climbing from 1 until they meet in the middle:
\( 1 \times 36 = 36 \), \( \; 2 \times 18 = 36 \), \( \; 3 \times 12 = 36 \), \( \; 4 \times 9 = 36 \), \( \; 6 \times 6 = 36 \).
5 is not a factor (\( 36 \div 5 \) leaves a remainder). At 6 the pair repeats itself, so we stop. Since 6 pairs with itself, it appears only once. The factors are \( \mathbf{1,\ 2,\ 3,\ 4,\ 6,\ 9,\ 12,\ 18,\ 36} \) — nine of them.
2. Is 29 prime or composite?
Check the small primes up to about \( \sqrt{29} \approx 5.4 \), so we only need to test 2, 3, and 5.
- Even? No — 29 is odd, so 2 is not a factor.
- Divisible by 3? The digits \( 2 + 9 = 11 \) is not a multiple of 3, so no.
- Divisible by 5? It doesn't end in 0 or 5, so no.
No small prime divides it, so its only factors are 1 and 29. Therefore 29 is prime.
3. Write 84 as a product of primes.
Build a factor tree, splitting until every tip is prime:
- \( 84 = 4 \times 21 \).
- \( 4 = 2 \times 2 \) and \( 21 = 3 \times 7 \).
- The tips 2, 2, 3, 7 are all prime, so we're done.
Collecting and grouping the repeat: \( 84 = 2 \times 2 \times 3 \times 7 = \mathbf{2^2 \times 3 \times 7} \). Check: \( 4 \times 3 \times 7 = 4 \times 21 = 84 \). ✓