Can somebody please explain how to use the Möbius function to solve this problem. https://www.hackerrank.com/contests/w3/challenges/gcd-product
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Can somebody please explain how to use the Möbius function to solve this problem. https://www.hackerrank.com/contests/w3/challenges/gcd-product
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Let f(n,m) denote the number of pairs (x, y), such that x ≤ n, y ≤ m, and gcd(x,y) = 1. WLOG, assume n ≥ m
Using inclusion exclusion,
If we store an array of prefix sums of mobius function, then f(n, m) can be calculated in .
Now let G(g, n, m) be the number of pairs (x, y), such that x ≤ n, y ≤ m, and gcd(x, y) = g.
Clearly, . There are clearly different values of this for all the g's
Our required answer is . Considering the different ranges in which G(g, n, m) is same, this can be calculated in O(n).
Thanks for the nice explanation.
I solved it little differently. After the step
Now I define a function f, such that for any prime p, f(pa) = p else f(n) = 1. It must be noted that
, Now If I substitute this expression in above expression and rearrange the multiplication we get
which reduces to
. Now this is very simple to evaluate.