Explain what is happening when using the difference method for determining the greatest common factor. Why does this work?

#color(blue)("The numeric reference")#

Let one of the common factors be #f=8#
let a numeric count be #n#

As the numbers to be tested I chose:

#8xx20=160#
#8xx15=120#
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#color(blue)("The underlying principle")#

As the process is based on subtraction then the starting point of

#160-120# has to have a difference that is related to one of the factors. In that: #" "120+nxx"some factor of 160"=160#

This will be true of every subtraction in that the difference will a factor.

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#color(blue)("The demonstration of process")#

Set the following
#8xx20=160 = 20f#
#8xx15=120=15f#

The subtraction process

#20f-15f=color(white)(1)5flarr" largest - smallest: next use the 15 & 5"#

#15f-color(white)(1)5f=10flarr" largest - smallest: next use the 10 & 5"#

#10f-color(white)(1)5f=color(white)(1)5flarr" largest - smallest: next use the 5 & 5"#

#5f-5f=0 larr" we have to stop at this point"#

This system is stating that the #GCF = 5f = 5xx8=40#
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#color(brown)("Numeric equivalent")#

#160-120=40" ......." ->color(white)(.) 5f->color(white)(.)5xx8=40#
#120-40=80" ........." ->10f->10xx8=80#
#80-40=40" ..........."->color(white)(.) 5f->color(white)(.)5xx8=40#
#40-40=0#

#GCF = 40#
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#color(blue)("Using prime factor trees")#

#GCF=2xx2xx2xx5=40#