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Let the numbers be $a, b$.
The $lcm(a,b)$ is given. Thus, you can find the product of the numbers using the formula: $ab=lcm(a,b)*gcd(a,b)$.
$(a+b)$ is also given. Now, you can easily solve these 2 equations.
For this particular question,
$a+b=20$ ----------(1)
$gcf(20,24)=4$
$ab=24*4=96$ ----------(2)
Let the numbers be $a,b$ and their $\gcd=d$,
Then $a=pd, b=qd$, where $p,q$ are co-prime.
$a+b=d(p+q)$
$lcm(a,b)=d(pq)$
Now, because $p, q$ are co-primes, $p+q$ and $pq$ will be co-prime too. i.e., $gcd(p+q,pq)=1$. Visit this question for the explanation.
Thus, $\gcd(d(p+q),d(pq))=d$
$\implies$ $\gcd((a+b),lcm(a,b))=d=\gcd(a,b)$
I am not experienced in writing mathematics symbol markdowns. I tried my best. Please edit if required.
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