#P12620. [NAC 2025] A Totient Quotient

[NAC 2025] A Totient Quotient

题目描述

For a positive integer kk, Euler's totient function ϕ(k)\phi(k) is defined as the number of positive integers less than or equal to kk and relatively prime to kk. For example, ϕ(9)=6\phi(9) = 6, ϕ(24)=8,\phi(24) = 8, and ϕ(1)=1\phi(1) = 1. (As a reminder, the greatest common divisor (gcd) of two positive integers aa and bb is the greatest positive integer that divides both aa and bb. Two positive integers are relatively prime if their gcd is 11.)

Euler's product formula gives the value of ϕ(k)\phi(k) in terms of the prime factorization of kk. For a prime pp, let νp(k)\nu_p(k) be the highest power of pp which divides kk (so for example, ν2(48)=4\nu_2(48) = 4, ν3(48)=1\nu_3(48)=1, and ν5(48)=0\nu_5(48)=0). If kk is a product of powers of prime factors, k=i=1jpiνpi(k)k = \prod_{i=1}^j p_i^{\nu_{p_i}(k)} (where the product only includes primes pip_i with νpi(k)>0\nu_{p_i}(k) > 0), 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 ϕ(m2)/ϕ(n2)\phi(m^2)/\phi(n^2), 128(2), 2021) proved the following fact about totient quotients: for any pair of positive integers aa, bb there is a unique pair of positive integers mm, nn for which:

  • ab=ϕ(m2)ϕ(n2);\frac{a}{b} = \frac{\phi(m^2)}{\phi(n^2)};
  • if a prime pp divides the product mnmn, then νp(m)νp(n)\nu_p(m) \neq \nu_{p}(n);
  • gcd(m,n)\gcd(m,n) is square-free: that is, for every prime pp, gcd(m,n)\gcd(m,n) is not divisible by p2p^2.

Conditions 2 and 3 guarantee that mm and nn 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 aa and bb and outputs the corresponding pair m,nm, n.

输入格式

The only line of input contains two space-separated integers aa and bb (1a,b10000 1 \leq a, b \leq 10\,000). These two integers are guaranteed to be relatively prime. Additionally, aa and bb will be chosen so that output values mm and nn are less than 2632^{63}.

输出格式

Print the two positive integers mm and nn satisfying all three of the conditions of The American Mathematical Monthly's theorem, separated by a space. It is guaranteed that m,n<263 m, n < 2^{63}.

9 13
18 13
19 47
13110 18612