I was wondering if its possible in constant time to calculate the above mentioned goal. I need it to solve a problem on codechef.
How we can find GCD(k + a, k + b) if we already know the GCD(a, b)?
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It is impossible to compute
gcd(a+k,b+k)in constant time knowing onlygcd(a,b).Suppose that
c,dare any two natural numbers withd < c.Let
Then we know in
O(1)time thatIf we could compute
gcd(a+k,b+k) = gcd(c,d)inO(1)additional time, then we could compute all gcds inO(1)time, which is impossible.Having said all that, it is of course possible that in some cases of interest, knowledge of
gcd(a,b)could lead to faster computation ofgcd(a+k,b+k)than would otherwise be possible.