A number theory problem by Christopher Boo

Number Theory Level 3

I have 2 positive integers. The greatest number that can divide both these numbers is 10, while the smallest positive number that is divisible by both is 100.

What is the product of these 2 integers?

1000 100 100000 10000

1 solution

Áron Bán-Szabó
Jun 10, 2017

Let $x$ , $y$ are the two numbers. We know that $\text{gcd}(x, y)=10$ and $\text{lcm}(x, y)=100$ . We will use one of the relationships between GCD and LCM: $\frac{xy}{\text{lcm}(x, y)}=\text{gcd}(x, y)$ . From that we get $\frac{xy}{100}=10$ , so $xy=10*100=\boxed{1000}$ .

We can check the solution with $x=20$ and $y=50$ .

A number theory problem by Christopher Boo

1 solution

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