Question

How many composite factors does 84 have?

Answer 1

`8` composite factors.

Explanation:

We would need to list all the factors of `84` first.

This is easier than it sounds. Factors are always in pairs, so if you know one of the pair you can divide that factor into `84` to find the other,

Half of the factors are less than `sqrt84` and are paired with factors greater than `sqrt84`

`sqrt84 = 9. ....`

Consider the numbers from `1" to " 9` .Which are factors?

`1," "2," "3," "4," "6," "7" "` are all factors.

Find the matching factors:

`1," "2," "3," "4," "6," "7," "12," "14," "21," "28," "42," "84`
`color(white)(xxxxxxxxxxxxxxxxxx)uarr`
`color(white)(xxxxxxxxxxxxxxxxx)sqrt84`

There are `12` factors. Of those, `1," "2," "3," "7` are not composite because they have less than `3` factors.

Therefore there are `12-4=8` composite factors

Answer 2

8.
The factors are 4, 6, 12, 14, 21, 28, 42, 84

Explanation:

Given `84=2^2*3*7`

84 has a few factors, as shown:

`84`
`=1*84`
`=2*42`
`=3*28`
`=4*21`
`=6*14`
`=7*12`

Excluding `1` and the prime factors `2`, `3`, `7`,
The number of composite factors `=12-4`
`=8`

Method 2:
Total number of composite factors
=total number of factors - total number of prime and unclassified factors
`=(2+1)(1+1)(1+1)-1-3`
`=12-4`
`=8`

Note: Total number of factors can be counted by multiplying the values of one added to the power of the primes.
i.e. Number of factors of `a^xb^yc^z`
`=(x+1)(y+1)(z+1)`