Logarithms

Algebra Level 4

A cool number is defined as a number NN that satisfies both of the following equations simultaneously.log3N=4+β1log5N=2+β2\log_3 N = 4 + \beta_1 \\ \log_5 N = 2 + \beta_2β1\beta_1 and β2\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)[0,1).

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