Question

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

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

Answer 1

See the explanation

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.

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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`
.......................................................................................................
`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`