✖️ Product, Pairs & Multiples

For any two numbers $a$ and $b$: $\text{HCF}(a,b)\times\text{LCM}(a,b) = a\times b$. Rearrange it to find whatever is missing — a number, the HCF, or the LCM.

• Why it works: write $a = h\cdot p$ and $b = h\cdot q$ where $h$ is the HCF and $p,q$ are co-prime. Then $\text{LCM} = h\cdot p\cdot q$, so $\text{HCF}\times\text{LCM} = h\cdot(h\,p\,q) = (hp)(hq) = a\,b$.

• Counting pairs: from a product, $p\,q = \dfrac{\text{product}}{h^2}$; from a sum, $p + q = \dfrac{\text{sum}}{h}$. Count only the co-prime splits — a shared factor would push the real HCF above $h$.

• Warning: the product rule holds for two numbers only — never for three.