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

Question

Use a numeric reference to compare and check the presented logic.

Luna Angel · Accepted Answer

See the explanation

#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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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#
.......................................................................................................
#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#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Using prime factor trees")#

#GCF=2xx2xx2xx5=40#

Autumn Arnold · Answer

undefined When using the difference method for determining the greatest common factor (GCF), you start by subtracting the smaller number from the larger number. Then, you repeat this process, replacing the larger number with the result of the subtraction and keeping the smaller number the same until both numbers are equal. The final result is the greatest common factor of the original two numbers.

This method works because it exploits the fact that the GCF of two numbers divides their difference. By iteratively subtracting the smaller number from the larger number, you're essentially reducing the problem to finding the GCF of two smaller numbers. Since the GCF divides both numbers, it also divides their difference. This process continues until both numbers are equal, which means the difference between them is zero, and the greatest common factor has been found.