Can a least common multiple be found for any two natural numbers?

Answer 1

Short answer: Yes.

Long answer:

Explanation:

lcm means least common multiple,

gcd means greatest common divisor,

|a| means the absolute of a,

a / b means a divided by b,

a * b means a multiplied by b

Premise: Let a and b be a natural numbers, i.e. a ⋲ IN, b ⋲ IN.

1: It is known that lcm(a, b) = (|a| * |b|) / gcd(a, b)

2: Also the gcd of two numbers is at least 1, or in math: ∀ a ⋲ IN: gcd(a, b) >= 1.

3: From 1 and 2 we can conclude: lcm(a, b) = |a| * |b| / gcd(a, b) <= |a| * |b|

4: From 3 and the premise we can conclude (because ∀ a ⋲ IN: |a| = a): lcm(a, b) <= a * b

5: Now the product of two natural numbers (like a and b) is a natural number as well, or in math: ∀ a ⋲ IN, b ⋲ IN: a * b ⋲ IN

6: From 2 and 5 we can finally conclude, that: ∀ a ⋲ IN, b ⋲ IN ∃ c ⋲ IN: lcm(a, b) <= c