Logarithms

Algebra Level 4

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 \([0,1)\).