Prime factorization and divisors

#prime factorization#divisors#exponents

Statement

Factor $5040$ and find the number of its divisors.

Hint

$5040=7!$ . Use $\tau(n)=\prod(\alpha_i+1)$ .

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Factoring. $5040=7!=2^4\cdot3^2\cdot5\cdot7$ .

Number of divisors.
$\tau(5040)=(4+1)(2+1)(1+1)(1+1)=5\cdot3\cdot2\cdot2=60.$

Conclusion. $5040=2^4\cdot3^2\cdot5\cdot7$ has $60$ positive divisors.