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Composing into number of integers with lower order

Source: Iranian third number theory finals problem 3

August 15, 2019

number theory

Problem Statement

Let

a,m

be positive integers such that

Ord_m (a)

is odd and for any integers

x,y

so that
1.

xy \equiv a \pmod m

Ord_m(x) \le Ord_m(a)

Ord_m(y) \le Ord_m(a)

We have either

Ord_m(x)|Ord_m(a)

Ord_m(y)|Ord_m(a)

.prove that

Ord_m(a)

contains at most one prime factor.