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gcd(a,b) = 1,

Since lcm is the multiple of a and b,

a|lcm(a,b) =⇒ lcm(a,b) = ax

b|lcm(a,b) =⇒ b|ax =⇒ ax = bq for q∈Z

Since gcd(a,b) = 1,b |x and b≤x =⇒ ab ≤ ax ---→ (O1)

However, ax is the least common multiple and ab is a common multiple of

a and b,

ax ≤ ab ---→ (O2)

by (O1) and (O2) ,

ax = ab

lcm(a,b) = ab

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