Positive Integral Factors

Rewrite as a prime factorization: $240={ 2 }^{ 4 }\times { 3 }^{ 1 }\times { 5 }^{ 1 }$ .

Thus some of the factors include:

${ 2 }^{ 0 },{ 2 }^{ 1 },{ 2 }^{ 2 },{ 2 }^{ 3 },{ 2 }^{ 4 }$

$3^{ 0 },{ 3 }^{ 1 }$

$5^{ 0 },{ 5 }^{ 1 }$

Thus, the total number of integral factors is the number of ways you could choose one from each list:

$5\times 2\times 2=20$

Notice that you can just add 1 to each of the exponents and then multiply them