Primes and Multiplication

1000ms

262144K

Description:

Let's introduce some definitions that will be needed later.

Let $$$prime(x)$$$ be the set of prime divisors of $$$x$$$. For example, $$$prime(140) = \{ 2, 5, 7 \}$$$, $$$prime(169) = \{ 13 \}$$$.

Let $$$g(x, p)$$$ be the maximum possible integer $$$p^k$$$ where $$$k$$$ is an integer such that $$$x$$$ is divisible by $$$p^k$$$. For example:

$$$g(45, 3) = 9$$$ ($$$45$$$ is divisible by $$$3^2=9$$$ but not divisible by $$$3^3=27$$$),
$$$g(63, 7) = 7$$$ ($$$63$$$ is divisible by $$$7^1=7$$$ but not divisible by $$$7^2=49$$$).

Let $$$f(x, y)$$$ be the product of $$$g(y, p)$$$ for all $$$p$$$ in $$$prime(x)$$$. For example:

$$$f(30, 70) = g(70, 2) \cdot g(70, 3) \cdot g(70, 5) = 2^1 \cdot 3^0 \cdot 5^1 = 10$$$,
$$$f(525, 63) = g(63, 3) \cdot g(63, 5) \cdot g(63, 7) = 3^2 \cdot 5^0 \cdot 7^1 = 63$$$.

You have integers $$$x$$$ and $$$n$$$. Calculate $$$f(x, 1) \cdot f(x, 2) \cdot \ldots \cdot f(x, n) \bmod{(10^{9} + 7)}$$$.

The only line contains integers $$$x$$$ and $$$n$$$ ($$$2 \le x \le 10^{9}$$$, $$$1 \le n \le 10^{18}$$$) — the numbers used in formula.

Print the answer.

10 2

20190929 1605

363165664

947 987654321987654321

593574252

In the first example, $$$f(10, 1) = g(1, 2) \cdot g(1, 5) = 1$$$, $$$f(10, 2) = g(2, 2) \cdot g(2, 5) = 2$$$.

In the second example, actual value of formula is approximately $$$1.597 \cdot 10^{171}$$$. Make sure you print the answer modulo $$$(10^{9} + 7)$$$.

In the third example, be careful about overflow issue.

Informação

Provedor Codeforces

Código CF1228C