Primary Divisonal 2012/4

Problem for Primary Group from Divisional Mathematical Olympiad will be solved here.
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Phlembac Adib Hasan
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Primary Divisonal 2012/4

Unread post by Phlembac Adib Hasan » Thu Dec 11, 2014 1:04 pm

The LCM of two numbers is $7$ times of their GCD. If the sum of the numbers is $56$, find their GCD.
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tanmoy
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Re: Primary Divisonal 2012/4

Unread post by tanmoy » Thu Dec 11, 2014 1:29 pm

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,$x=7$ :)
"Questions we can't answer are far better than answers we can't question"

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