Gcd wanted
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Raiyan Jamil
- Posts:138
- Joined:Fri Mar 29, 2013 3:49 pm
The sum of two numbers is 392 . Their LCM is 7 times their GCD . What is the value of their GCD and why ?
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Re: Gcd wanted
let, $\gcd(a,b)=x\Rightarrow a=xa_{1}; b=xb_{1}$ where $(a_{1},b_{1})=1$
and given $\gcd(a,b)*7=lcm[a,b]\Leftrightarrow 7x=lcm[a,b]$
so,
$[a,b]=[xa_{1},xb_{1}]=xa_{1}b_{1}=7x$
$\Rightarrow 7x=xa_{1}b_{1}$
$\Rightarrow a_{1}b_{1}=7$
$\therefore a_{1}=1 ; b_{1}=7$
$a=xa_{1}=x*1=x ; b=xb_{1}=7*x=7x$
and given $a+b=392\Rightarrow x+7x=392\Rightarrow 8x=392\Rightarrow x=49$
so gcd is 49 [ans]
and given $\gcd(a,b)*7=lcm[a,b]\Leftrightarrow 7x=lcm[a,b]$
so,
$[a,b]=[xa_{1},xb_{1}]=xa_{1}b_{1}=7x$
$\Rightarrow 7x=xa_{1}b_{1}$
$\Rightarrow a_{1}b_{1}=7$
$\therefore a_{1}=1 ; b_{1}=7$
$a=xa_{1}=x*1=x ; b=xb_{1}=7*x=7x$
and given $a+b=392\Rightarrow x+7x=392\Rightarrow 8x=392\Rightarrow x=49$
so gcd is 49 [ans]