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Phi function and number of divisors

Source: Iran Team selection test 2024 - P5

May 19, 2024
number theory

Problem Statement

Suppose that we have two natural numbers x,y≤100!x , y \le 100! with undetermined values. Prove that there exist natural numbers m,nm , n such that values of x,yx , y get uniquely determined according to value of φ(d(my))+d(φ(nx))\varphi(d(my))+d(\varphi(nx)). ( for each natural number nn , d(n)d(n) is number of its positive divisors and φ(n)\varphi(n) is the number of the numbers less that nn which are relatively prime to nn. )
Proposed by Mehran Talaei
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