In number theory, a multiplicative function is an arithmetic function F(n)
of the positive integer n with property that F(1) = 1 and whenever a and b
are coprime (gcd(a, b) = 1), then F(ab) = F(a)F(b).
The function E(n) defined by E(n) = 1 if n = 1 and = 0 if n > 1, is sometimes
called multiplication unit for Dirichlet convolution or simply the unit function.
If F and G are two multiplicative functions, one defines a new multiplicative
* G, the Dirichlet convolution of F and G, by
where the sum extends over all positive divisors d
. With this operation,
the set of all multiplicative functions turns into an abelian group; the identity element is E
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In this task you have to find the inverse of a multiplicative function.
To cope with overflow problem, we define arithmetic functions as:
F: N —> Z2007
where N is the set of positive integers, and Z2007
is a residue ring (ring of integers 0–2006, where arithmetic operations + and × are performed
modulo 2007). Function G is called the inverse of function F if and only if F
* G = G
* F = E.
You are given the first N values of function F, you need to find the first N values of
the inverse function G.
In the first line of the input one number N is written (1 ≤ N ≤ 104).
In the second line values F(1), F(2), F(3), …, F(N) are listed. Numbers are
separated by spaces. (Each value is nonnegative and doesn't exceed 2006.)
In the first line of the output print first N values of inverse function G,
separated by spaces: G(1), G(2), …, G(N).
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2006 2006 0 2006 1 2006 0 0 1 2006 0 2006 1 1 0
Problem Source: Novosibirsk SU Contest. Petrozavodsk training camp, September 2007