MathDB

1

Part of 2008 239 Open Mathematical Olympiad

Problems(2)

Ascending proper divisors

Source: 239 2008 S1

7/28/2020
Composite numbers aa and bb have equal number of divisors. All proper divisors of aa were written in ascending order and all proper divisors of bb were written under them in ascending order, then the numbers that are below each other were added together. It turned out that the resulting numbers formed a set of all proper divisors of a certain number. What are the smallest values that aa and bb take?
Proper divisors set

Source: 239 2008 J1

7/28/2020
An odd natural number kk is given. Consider a composite number nn. We define d(n)d(n) the set of proper divisors of number nn. If for some number mm, d(m)d(m) is equal to d(n)∪{k}d(n) \cup \{ k \}, we call nn a good number. prove that there exist only finitely many good numbers. (A proper divisor of a number is any divisor other than one and the number itself.)