MISSION · Number sense
Factors, Multiples, and Primes
Every number has one recipe, and it explains a lot
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Every whole number above 1 is a product of primes in exactly one way, and that recipe answers most questions you can ask about factors and multiples.
Factors and multiples point opposite ways
A factor divides into a number exactly: the factors of 12 are 1, 2, 3, 4, 6, 12. A multiple is what you get by multiplying it: 12, 24, 36, 48, and on forever.
So factors are finite and live below the number; multiples are infinite and live above it. Mixing up the words is the most common source of a wrong answer in this topic, and the directions are worth fixing in mind before anything else.
Primes are the building blocks
A prime has exactly two factors: itself and 1. That’s 2, 3, 5, 7, 11, 13, and so on.
Two clarifications people trip on. 1 is not prime — it has only one factor, and if it counted, every number would have endless recipes (\(6 = 2 \times 3 = 1 \times 2 \times 3 = \ldots\)), wrecking the uniqueness that makes primes useful. And 2 is prime, the only even one — every other even number has 2 as a third factor.
The unique recipe
Break a number down until only primes remain, and you always land on the same set:
\[60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\]Start with \(6 \times 10\) or \(4 \times 15\) or \(2 \times 30\) — the route differs, the destination doesn’t. That’s the fundamental theorem of arithmetic, and it’s the reason primes are called the atoms of the number system.
Every factor of 60 is some combination of those primes, which is why the recipe answers so much at once.
HCF and LCM from the recipe
Write both numbers as primes, then:
- Highest common factor — take the primes they share, at the lower power.
- Lowest common multiple — take every prime that appears, at the higher power.
For 12 and 18: \(12 = 2^2 \times 3\) and \(18 = 2 \times 3^2\).
HCF: shared primes at the lower power, \(2 \times 3 = 6\). LCM: all primes at the higher power, \(2^2 \times 3^2 = 36\).
A check worth knowing: \(\text{HCF} \times \text{LCM}\) always equals the product of the two numbers. Here \(6 \times 36 = 216 = 12 \times 18\).
Where you actually use them
HCF splits things into the largest equal groups — the biggest square tile that fits a room exactly, or the largest identical party bags from two different sweet counts.
LCM finds when cycles coincide — two buses leaving every 12 and 18 minutes next depart together after 36. It’s also what a common denominator is: the LCM of the two bottoms.
Common mix-ups
- Swapping HCF and LCM. The common factor is smaller than both numbers; the common multiple is bigger.
- Calling 1 prime, or forgetting 2 is.
- Stopping the factor tree early, leaving a composite like 4 in the answer.
- Listing multiples to find an LCM when the numbers are large. The prime method is faster and doesn’t run out of paper.
Check yourself
Write 84 as a product of primes, then find the HCF and LCM of 84 and 60.
\(84 = 2^2 \times 3 \times 7\) and \(60 = 2^2 \times 3 \times 5\). HCF is \(2^2 \times 3 = 12\); LCM is \(2^2 \times 3 \times 5 \times 7 = 420\).
LEVEL 3 · REFLECT
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