One set of primes, HCF and LCM
Split 180 two ways. You get the same primes.
- 180 splits two ways: 4 × 45, and 6 × 30.
- Left tree: 4 opens to 2 × 2.
- Left tree: 45 opens to 3 × 3 × 5.
- Right tree: 6 opens to 2 × 3, and 30 opens to 2 × 3 × 5.
- Both trees end on the same primes: 2 × 2 × 3 × 3 × 5.
The rule
Every composite number splits into primes in only one way. Only the order can change. That fact is the Fundamental Theorem of Arithmetic.
HCF and LCM from primes
A power says how many times a prime appears. 2² is 2 × 2.
The HCF is the highest common factor. Take each shared prime with its smallest power.
The LCM is the lowest common multiple. Take every prime with its greatest power.
| Answer | Primes used | Value |
|---|---|---|
| HCF | 2² × 3 | 12 |
| LCM | 2² × 3² × 5 | 180 |
For 3, the powers are 3² and 3¹. The HCF takes 3¹. The LCM takes 3².
5 is in only one number. The HCF skips it. The LCM takes it.
HCF × LCM = product of the two numbers.
Check: 12 × 180 = 2160, and 36 × 60 = 2160.
This is true for two numbers. It can fail for three numbers. Take 4, 6 and 8. The HCF is 2 and the LCM is 24. But 2 × 24 = 48, and 4 × 6 × 8 = 192.
Two more uses
- Two runners take 15 and 20 minutes for one round. They start together. They meet at the start after LCM(15, 20) minutes. 15 = 3 × 5 and 20 = 2² × 5. So the LCM is 2² × 3 × 5 = 60 minutes.
- Can 15ⁿ end in 0, for a whole number n above 0? 15ⁿ = 3ⁿ × 5ⁿ. Its only primes are 3 and 5. A number ending in 0 is a multiple of 10. So 2 is a factor. So 15ⁿ never ends in 0.
Now try it. Pick two numbers. Read the HCF and LCM off their primes.
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