How can you simplify the following fraction using prime factorization?: #456/939#

1 Answer
May 6, 2016

#456/939 = 152/313#

Explanation:

We can factor #456# as expressed by this factor tree:

#color(white)(00000)456#
#color(white)(0000)"/"color(white)(000)"\"#
#color(white)(000)2color(white)(0000)228#
#color(white)(0000000)"/"color(white)(000)"\"#
#color(white)(000000)2color(white)(0000)114#
#color(white)(0000000000)"/"color(white)(000)"\"#
#color(white)(000000000)2color(white)(0000)57#
#color(white)(0000000000000)"/"color(white)(00)"\"#
#color(white)(000000000000)3color(white)(000)19#

We can factor #939# as expressed by this factor tree:

#color(white)(00000)939#
#color(white)(0000)"/"color(white)(000)"\"#
#color(white)(000)3color(white)(0000)313#

It might take us a while to find that #313# is prime, but we don't actually need to. The only factors we need to check for are those common with #456#, i.e. #2#, #3# and #19#.

So the only common factor is #3# (once) and we find:

#456/939 = (color(red)(cancel(color(black)(3)))xx152)/(color(red)(cancel(color(black)(3)))xx313) = 152/313#

#color(white)()#
Alternative method

Rather than doing all of this factoring, we can find the GCF of #456# and #939# using the following method:

  • Divide the larger number by the smaller to give a quotient and remainder.

  • If the remainder is #0# then the smaller number is the GCF.

  • Otherwise repeat with the smaller number and the remainder.

So in our example:

#939/456 = 2# with remainder #27#

#456/27 = 16# with remainder #24#

#27/24 = 1# with remainder #3#

#24/3 = 8# with remainder #0#

So the GCF is #3# and we can write:

#456/939 = ((456/3))/((939/3)) = 152/313#