A number theory problem by Christopher Boo

Number Theory Level 1

True or False?

$x$ and $y$ are two positive integers.
If $\text{gcd}{(x, y)} = x,$ then $\text{lcm}{(x, y)} = y.$

Notations: $\gcd(\cdot)$ and $\text{lcm}(\cdot)$ denote the greatest common divisor function and the lowest common multiple function, respectively.

True False

1 solution

Tapas Mazumdar
Jun 10, 2017

If $\gcd (x,y) = x$ then it means that $y$ can be represented as $nx$ for some positive integer $n$ . So, $y$ is an integer multiple of $x$ and since $\text{lcm} (x,nx) = nx$ , so we get $\text{lcm} (x,y) = y$ .

A number theory problem by Christopher Boo

1 solution

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