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Suppose that we have two integers $a$ and $b$. Now say that $G = \gcd(a,b)$ and $L = \mathrm{lcm}(a,b)$. Now the value of $G$ and $L$ is given and another integer $c$'s value is given. How can we find $\gcd(a+c,b+c)$ and $\mathrm{lcm}(a+c,b+c)$ from $G$, $L$ and $c$?

What if we have $n$ arbitrary numbers. I know the GCD and LCM of those numbers but not the actual values of those numbers. Now I want to add $c$ with all of those numbers, what will be the new GCD, LCM of those numbers?

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    Do your variables $c$ and $C$ refer to the same number? In that case please consistently use the same letter.2012-10-22
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    The gcd(a+c,b+c) is a divisor of $a-b$. Therefore you have a finite number of possibilities as $c$ varies.2012-10-22
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    I believe there is a theorem (a special case of this) about Mersenne numbers which states that $\gcd ({2^n} - 1,{2^m} - 1) = {2^{\gcd (n,m)}} - 1$.2012-10-22

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