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Baltic Way 1992 · Problem 2

Number Theory

Denote by d(n)d(n) the number of all positive divisors of a positive integer nn (including 1 and nn ). Prove that there are infinitely many nn such that nd(n)\frac{n}{d(n)} is an integer.

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Topics

Primes · Divisibility and factorization · Arithmetic functions

Solutions

Solution

Solution:

Consider numbers of the form ppn−1p^{p^{n}-1} where pp is an arbitrary prime number and n=1,2,…n=1,2, \ldots

Contest context

Results from Baltic Way 1992

8 teams

Mean score
4.4 / 5
Scores of 4 or 5
7 / 8
Estonia
5 / 5

Score distribution

01
10
20
30
40
57
All team scores
TeamScore
Denmark5 / 5
St. Petersburg5 / 5
Poland5 / 5
Latvia5 / 5
Iceland0 / 5
Lithuania5 / 5
Estonia5 / 5
Sweden5 / 5