HCF & LCM

At a glance 12 formulas

1. 1Prime-power methodFactorise each number. HCF takes the lowest power of each common prime; LCM takes the highest power of every prime.
2. 2HCF or LCM?Greatest size, largest tile, equal groups with nothing left over: HCF. Least number, first time together, ringing again: LCM.
3. 3HCF times LCM$\text{HCF} \times \text{LCM} = a \times b$$\text{for two numbers only}$
4. 4Numbers from the HCF$a = hm,\ b = hn,\ \text{LCM} = hmn$$m \text{ and } n \text{ co-prime}$
5. 5Special pairsCo-prime numbers: HCF 1, LCM the product. If one number divides the other, the HCF is the smaller and the LCM the larger.
6. 6Fractions, lowest terms$\text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$$\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}$
7. 7Least number, same remainder$N = \text{LCM} \times k + r$
8. 8Least number, same shortfall$N = \text{LCM} \times k - g$$g = \text{divisor} - \text{remainder, the same for all}$
9. 9Greatest divisor, same remainder$\text{HCF}(b - a,\ c - b)$$\text{remainders } r_1, r_2 \text{ given: } \text{HCF}(a - r_1,\ b - r_2)$
10. 10Euclid's shortcut$\text{HCF}(a, b) = \text{HCF}(a,\ b - a)$$\text{HCF}(1530, 1665) = \text{HCF}(1530, 135) = 45$
11. 11Counting times together$\text{times together in } T = \left\lfloor \dfrac{T}{\text{LCM}} \right\rfloor + 1$$\text{the } +1 \text{ counts the start}$
12. 12Sum and HCF givenWrite the numbers as $hm$ and $hn$ with $m + n = \tfrac{S}{h}$, then count only the co-prime splits of $\tfrac{S}{h}$.