# Logarithms

**Algebra**Level 5

A *cool number* is defined as a number \(N\) that satisfies both of the following equations simultaneously.\[\log_3 N = 4 + \beta_1 \\ \log_5 N = 2 + \beta_2\]\(\beta_1\) and \(\beta_2\) are fractional parts of the number. How many integral cool numbers are there?

**Note:** A fractional part lies in the interval \([0,1)\).