Problem #PRU-21989

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Numeral systems Decimal number system Euler's theorem Methods Pigeonhole principle Pigeonhole principle (other)
 12–14  3.0

Problem

Prove that there is a power of \(3\) that ends in \(001\). You can take the following fact as given: if the product \(a\times b\) of two numbers is divisible by another number \(c\), but \(a\) and \(c\) share no prime factors (we say that \(a\) and \(c\) are coprime) then \(b\) must be divisible by \(c\).