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 = \dfrac{\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 is a multiple of the HCF
\gcd(a, b) \mid \text{LCM}(a, b)
Ratio pair
\text{numbers} = hm,\ hn; \quad \text{LCM} = hmn
Finding HCF & LCM (incl. fractions and decimals)
HCF by factors
\text{HCF} = p_1^{\min} \times p_2^{\min} \times \cdots
common primes, lowest powers
LCM by factors
\text{LCM} = p_1^{\max} \times p_2^{\max} \times \cdots
all primes, highest powers
HCF of fractions
\text{HCF}\!\left(\dfrac{a}{b}, \dfrac{c}{d}\right) = \dfrac{\gcd(a, c)}{\text{LCM}(b, d)}
LCM of fractions
\text{LCM}\!\left(\dfrac{a}{b}, \dfrac{c}{d}\right) = \dfrac{\text{LCM}(a, c)}{\gcd(b, d)}
LCM from the HCF
\text{LCM}(a, b) = \dfrac{a \times b}{\gcd(a, b)}
Standard word problems (tiles, bells, groups, divisible numbers)
Same remainder r
N = \text{LCM}(d_1, d_2, \ldots) \times k + r
Remainder is divisor minus c
r_i = d_i - c \Rightarrow N = \text{LCM} \times k - c
Divides with remainders
\text{answer} = \gcd(a - r_1,\ b - r_2,\ c - r_3)
Largest tile count
\text{tiles} = \dfrac{L \times W}{h^2}, \quad h = \gcd(L, W)
Two-step LCM/HCF cases (extra condition, N-digit bounds)
Extra divisibility condition
N = Lk + r, \quad N \equiv 0 \pmod{p} \ \Rightarrow\ Lk \equiv -r \pmod{p}
Reconstruction from HCF and LCM
ab = \dfrac{\text{LCM}}{h}, \quad \gcd(a, b) = 1, \quad \text{numbers} = ha,\ hb
Pair count
\#\{(a, b): ab = M,\ \gcd(a, b) = 1,\ a \le b\}
Same unknown remainder
\text{answer} = \gcd(a - b,\ b - c,\ a - c)