How many composite factors does 84 have?

2 Answers
Apr 25, 2018

#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

Apr 25, 2018

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)#