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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Updated on July 26, 2022

Comments

  • azif00
    azif00 over 1 year

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

    • 19aksh
      19aksh over 4 years
      Welcome to MSE. Please show us what you've tried.
    • lulu
      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
      J. W. Tanner over 4 years
      and you need to consider only those that are multiples of $10$