Not knowing the $\mathrm{gcd}$ and $\mathrm{lcm}$ and knowing $\mathrm{gcd+lcm}$, how to find $a$ and $b$ in $\mathrm{gcd}(a,b)$?

Question

Here's what we have:

$\mathrm{gcd}(a,b)=d$ ; $\mathrm{lcm}(a,b)=m$ ; $a+b=30$ ; $m+d=42$ ; $b>a$.

What I tried:

if $d$ divides $a$ and $b$ so it divides $a+b$ so $d$ divides $30$. And with $\mathrm{gcd}$ and $\mathrm{lcm}$ rules I found that $md=ab$. But that's all, I didn't know how to continue.

Thanks.

Well, if all else fails a simple search is easy enough. There are only $14$ possible values for $a$, after all. — lulu, Aug 01 '20 at 16:05

score 3 · Accepted Answer · answered Aug 01 '20 at 16:18

3

Hint: This might help narrowing down the search: $d$ also divides $m$ (and hence it divides $42$).

answered Aug 01 '20 at 16:18

ir7

6,249

thanks, so d divides 12, but how to continue? – 18.99 Aug 01 '20 at 16:26
2

d divides 6 ( = gcd(30,42)). So, d can be 2, 3 or 6. You can study each case and see if you can find a and b. If d=2, then m=40, so ab=md=80. And so on. – ir7 Aug 01 '20 at 16:31
thanks a lot, that's logical. – 18.99 Aug 01 '20 at 16:37
another quick question, why d can't be equaled to 1? – 18.99 Aug 01 '20 at 17:04
It can. My bad. – ir7 Aug 01 '20 at 17:06
aaah ok, thanks – 18.99 Aug 01 '20 at 17:07

Not knowing the $\mathrm{gcd}$ and $\mathrm{lcm}$ and knowing $\mathrm{gcd+lcm}$, how to find $a$ and $b$ in $\mathrm{gcd}(a,b)$?

1 Answers1