Quant · Number System

HCF and LCM

2 sub-topics · 20 lessons · 63 questions · 9 PYQs

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HCF and LCM Basics10 lessons · 30 questions · 5 PYQs

HCF and LCM Applications10 lessons · 33 questions · 4 PYQs

Formula sheet

Count with a given gcd (Indore 2026)
$\#\{n\le N:\ \gcd(n,N)=d\}=\varphi(N/d)$
Same remainder with every divisor
$N=\text{LCM}(a,b,c)\cdot k+r$
Euler's phi (co-primes below N)
$\varphi(N)=N\prod_{p\mid N}\left(1-\tfrac1p\right)$
HCF of two numbers of the form (power of a) minus 1
$\text{HCF}(a^m-1,\ a^n-1)=a^{\gcd(m,n)}-1$
Numbers from HCF and co-prime ratio
$a=hp,\ b=hq,\ \gcd(p,q)=1\ \Rightarrow\ \text{LCM}=hpq$
Euclid: HCF from the difference
$\text{HCF}(a,b)=\text{HCF}(b,\ a-b)$
Product rule (two numbers)
$\text{HCF}\times\text{LCM}=a\times b$
Indore 2022 identity
$\frac{(a+b)^2-(a-b)^2}{4}=ab=\text{HCF}\times\text{LCM}$