How many factors, of the number 1080, are a multiple of 2?

Solution:

• Let us first apply prime factorization and express these numbers in terms of it prime factors. \(1080 = 2*2*2*3*3*3*5\)

o This can be written as:

 \(1080 = 2^3 * 3^3 * 5\)
• As mentioned before, any factors which are a multiple of \(2\), has to be even and hence we are simply looking for even factors of the number \(1080\).
• But how do we find the number of even factors?

o Per our conceptual knowledge, we know that a number can have either odd factors or even factors or both.
o Thus, the total factors of any number have to be a combination of only these two types of factors.

 Total factors = Even Factors + Odd factors
• Since we need even factors, we can re-arrange and write the formula as follows:
• Even factors = Total factors – Odd factors. …………..[I]
• Per our conceptual understanding,

o Total factors of \(1080\)= (Power of 2 +1) * (Power of 3+1) * (Power of 5+1)

 Total factors = \((3+1) * (3+1) * (1+1)\)
 Total factors = \(4 * 4 * 2 = 32\) factors………..[II]
• Total odd factors of \(1080\) can be found out using the powers of \(3\) and \(5\) only. we cannot include \(2\) as that would give us even factors

o So, odd factors of \(1080\) = (Power of 3+1) * (Power of 5 +1)

 Odd Factors = \((3+1) * (1+1)\)
 Odd Factors = \(4 * 2 = 8\) factors …………….[III]
• Substituting the values obtained in [II] and [III] in equation [I]we get:

o Even factors = Total Factors – Odd Factors

 Even factors = \(32- 8\)
 \(24\) factors