HCF & LCM
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HCF & LCM: definitions and core relations
Product relation (two numbers)
\text{HCF} \times \text{LCM} = a \times b
Other number
b = \frac{\text{HCF} \times \text{LCM}}{a}
Co-prime case
\gcd(a,b) = 1 \Rightarrow \text{LCM}(a,b) = ab
HCF divides every difference
\gcd(a,b) \mid (a-b)
LCM multiple of HCF
\gcd(a,b) \mid \text{LCM}(a,b)
Finding HCF & LCM (incl. fractions and decimals)
HCF by factors
\text{HCF} = p_1^{\min} \cdot p_2^{\min} \cdots \text{(common primes, lowest powers)}
LCM by factors
\text{LCM} = p_1^{\max} \cdot p_2^{\max} \cdots \text{(all primes, highest powers)}
HCF of fractions
\text{HCF}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\gcd(a, c)}{\text{LCM}(b, d)}
LCM of fractions
\text{LCM}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\gcd(b, d)}
Two-number division method
\text{LCM} = \frac{a \times b}{\gcd(a,b)}
Standard word problems (tiles, bells, groups, divisible numbers)
Same remainder r
N = \text{LCM}(d_1, d_2, \dots) \times k + r
Different remainders, N ≡ −c
N = \text{LCM} \times k - c \text{ when } r_i = d_i - c
Largest tile count
\text{tiles} = \frac{L \times W}{h^2},\ h = \gcd(L, W)
Greatest n-digit multiple
\text{answer} = \underbrace{99\ldots9}_{n} - \left(\underbrace{99\ldots9}_{n} \bmod \text{LCM}\right)
Two-step LCM/HCF cases (extra condition, N-digit bounds)
Extra divisibility condition
N = Lk + r,\ N \equiv 0 \pmod p \ \Rightarrow\ Lk \equiv -r \pmod p
Reconstruction
ab = \frac{\text{LCM}}{h},\quad \gcd(a, b) = 1
Pair count
\#\{(a,b): ab = M,\ \gcd(a,b)=1,\ a \le b\}