# Find all positive integers $a$ and $b$ satisfying $\gcd (a,b)=10$ and $\operatorname{lcm} (a,b)=100$ simultaneously. [closed]

Find all positive integers $$a$$ and $$b$$ satisfying $$\gcd (a,b)=10$$ and $$\operatorname{lcm} (a,b)=100$$ simultaneously.

## closed as off-topic by Adam Chalumeau, Ak19, lulu, John Omielan, max_zornJul 29 at 2:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Adam Chalumeau, Ak19, lulu, John Omielan, max_zorn
If this question can be reworded to fit the rules in the help center, please edit the question.

• Welcome to MSE. Please show us what you've tried. – Ak19 Jul 28 at 12:07
• $100$ only has nine divisors....if you can't think of anything else, just work with that list. – lulu Jul 28 at 12:09
• and you need to consider only those that are multiples of $10$ – J. W. Tanner Jul 28 at 12:23

From the given,

$$a=10n,b=10m$$ where $$n,m$$ are relative primes and $$10nm=100.$$

Hence from the factorizations of $$10$$, the solutions

$$10,100;20,50;50,20;100,10.$$

WLOG

$$\dfrac aA=\dfrac bB=10;(A,B)=1$$

$$[a,b]=[10A,10B]=10[A,B]=100$$

$$\implies[A,B]=?$$ with $$(A,B)=1$$