Find all positive integers $a$ and $b$ satisfying $\gcd (a,b)=10$ and $\operatorname{lcm} (a,b)=100$ simultaneously.
1,323
Solution 1
From the given,
$$a=10n,b=10m$$ where $n,m$ are relative primes and $$10nm=100.$$
Hence from the factorizations of $10$, the solutions
$$10,100;20,50;50,20;100,10.$$
Solution 2
WLOG
$\dfrac aA=\dfrac bB=10;(A,B)=1$
$[a,b]=[10A,10B]=10[A,B]=100$
$\implies[A,B]=?$ with $(A,B)=1$
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azif00
Updated on July 26, 2022Comments

azif00 over 1 year
Find all positive integers $a$ and $b$ satisfying $$\gcd (a,b)=10$$ and $$\operatorname{lcm} (a,b)=100$$ simultaneously.

19aksh over 4 yearsWelcome to MSE. Please show us what you've tried.

lulu over 4 years$100$ only has nine divisors....if you can't think of anything else, just work with that list.

J. W. Tanner over 4 yearsand you need to consider only those that are multiples of $10$
