#P12620. [NAC 2025] A Totient Quotient
[NAC 2025] A Totient Quotient
题目描述
For a positive integer , Euler's totient function is defined as the number of positive integers less than or equal to and relatively prime to . For example, , and . (As a reminder, the greatest common divisor (gcd) of two positive integers and is the greatest positive integer that divides both and . Two positive integers are relatively prime if their gcd is .)
Euler's product formula gives the value of in terms of the prime factorization of . For a prime , let be the highest power of which divides (so for example, , , and ). If is a product of powers of prime factors, (where the product only includes primes with ), then
$$\phi(k) = \prod_{i=1}^j \left[(p_i - 1)\left(p_i^{\nu_{p_i}(k)-1}\right)\right]. $$A recent edition of The American Mathematical Monthly (Li et al., Positive Rational Numbers of the Form , 128(2), 2021) proved the following fact about totient quotients: for any pair of positive integers , there is a unique pair of positive integers , for which:
- if a prime divides the product , then ;
- is square-free: that is, for every prime , is not divisible by .
Conditions 2 and 3 guarantee that and are the unique smallest pair of positive integers satisfying condition 1.
You'd like to verify this claim numerically. Write a program which takes as input two integers and and outputs the corresponding pair .
输入格式
The only line of input contains two space-separated integers and (). These two integers are guaranteed to be relatively prime. Additionally, and will be chosen so that output values and are less than .
输出格式
Print the two positive integers and satisfying all three of the conditions of The American Mathematical Monthly's theorem, separated by a space. It is guaranteed that .
9 13
18 13
19 47
13110 18612