Let a and b be two positive integers and and then the numbers a and b are

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Concept:

Let, gcd (a,b) = g

then,  integer m and n such that a = mg & b = ng where (m, n) = 1.

Now, lcm is the least common multiple means it is the product of common factors of two numbers (a and b) and the factors which are coprime to each other.

⇒lcm (a,b) = gmn

Calculations:

Let a and b be two positive integers.

Consider,  gcd (a,b) = g

then,  integer m and n such that a = mg & b = ng where (m, n) = 1.

Now, lcm is the least common multiple means it is the product of common factors of two numbers (a and b) and the factors which are coprime to each other.

⇒lcm (a,b) = gmn

Consider, gcd(a, b). lcm(a, b)

⇒ (g)(gmn)

⇒(gm)(gn)

⇒ ab

Hence, Let a and b be two positive integers. Then gcd(a, b). lcm(a, b) equals: ab

Suppose we have two positive integers $a$ and $b$ which satisfy the condition $a^3 − 2b^3 = 2$.

It then follows that the greatest common divisor of $a$ and $b$ must be either $1$ or $2$. True or false?

I tried solving by supposing $a$ and $b$ are composite numbers sharing $\beta$ as a common factor.

Letting $a = \alpha \beta$ and $b = \gamma \beta$, I factorised $\beta^3 (\alpha^3 + 2 \gamma^3) = 2$, $$\beta^3 = \frac{2}{\alpha^3 + 2 \gamma^3},$$ which is lesser than a whole number therefore both $a$ and $b$ must be coprime to each other, i.e $\gcd(a, b) = 1$ only.