Leave lesson

Number · Integers, powers & roots

1 / 9

HCF and LCM

What highest common factor and lowest common multiple actually mean, how prime factors give both (lowest powers vs highest powers), and how to spot which one a word problem is asking for.

Number · Integers, powers & roots

HCF and LCM

What highest common factor and lowest common multiple actually mean, how prime factors give both (lowest powers vs highest powers), and how to spot which one a word problem is asking for.

Why it works

Two numbers, two different questions:
  • HCF — the biggest number that divides into both. It's about breaking
the numbers down, so it's at most the smaller number.
  • LCM — the smallest number both divide into. It's about building up,
so it's at least the larger number.

Keeping those directions straight kills the classic swap: the HCF of 72 and 120 can't be 360, because 360 doesn't divide into either.

Prime factors give both by comparing atoms. With 72=23×3272 = 2^3 \times 3^2 and 120=23×3×5120 = 2^3 \times 3 \times 5:
  • HCF: take each shared prime to its LOWER power — what both numbers can
supply: 23×3=242^3 \times 3 = 24. (In a Venn diagram, the intersection.)
  • LCM: take every prime to its HIGHER power — enough to contain both:
23×32×5=3602^3 \times 3^2 \times 5 = 360. (The whole Venn diagram.)

Why lower and higher? A common factor can only use atoms both numbers own — the overlap. A common multiple must hold all of each number's atoms — the union. And a tidy check: HCF×LCM=24×360=25 920=72×120\text{HCF} \times \text{LCM} = 24 \times 360 = 25\,920 = 72 \times 120 — the product of the numbers, always, because overlap + union share out the atoms exactly once.

"Multiply them" is only sometimes the LCM. LCM=\text{LCM} = product only when the numbers share no factor (HCF 1). 24×40=96024 \times 40 = 960, but LCM(24,40)=120(24, 40) = 120 — the shared 232^3 mustn't be counted twice.

Word problems: which direction?
  • Events that repeat and re-align — buses leaving together, lights
flashing together — want the LCM of the cycles: the first time both counts land together. Buses every 24 and 40 minutes: LCM =120= 120 minutes.
  • Cutting or sharing into the largest equal pieces with nothing left over
wants the HCF: ribbons of 84 cm and 60 cm cut into equal longest pieces — HCF =12= 12 cm.

Ask: "am I building up to a shared future moment (LCM), or cutting down to a shared size (HCF)?"