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Primary Divisonal 2012/4

Posted: Thu Dec 11, 2014 1:04 pm
by Phlembac Adib Hasan
The LCM of two numbers is $7$ times of their GCD. If the sum of the numbers is $56$, find their GCD.

Re: Primary Divisonal 2012/4

Posted: Thu Dec 11, 2014 1:29 pm
by tanmoy
The GCD is $7$.
Soppose,the numbers are $a$ and $b$ and suppose the GCD of the numbers are $x$.So,$a=xa_{1}$ and $b=xb_{1}$ for some integers $a_{1}$ and $b_{1}$ so that the GCD of $a_{1}$ and $b_{1}$ is $1$
$\therefore$ $7x=xa_{1}b_{1}$.Or,$a_{1}b_{1}=7$.So,one of $a_{1}$ and $b_{1}$ is $1$ and other is $7$.
Now,$xa_{1}+xb_{1}=56$
Or,$x(a_{1}+b_{1})=56$
Or,$8x=56$
Or,$x=7$ :)