Question

True or False ? If `2` divides `gcf(a,b)` and `2` divides `gcf(b,c)` then `2` divides `gcf(a,c)`

Hi i need to answer this true or false question. If it's true I need to prove it but if it's false I need a counter example.

Answer 1

Please see below.

Explanation:

GCF of two numbers, say `x` and `y`, (in fact even more) is a common factor, which divides all the numbers. We write it as `gcf(x,y)`. However, note that GCF is greatest common factor and every factor of these numbers, is a factor of GCF too.

Also note that if `z` is a factor of `y` and `y` is a factor of `x`, then `z` is a factor o `x` too.

Now as `2` divides `gcf(a,b)`, it means, `2` divides `a` and `b` too and therefore `a` and `b` are even.

Similarly, as `2` divides `gcf(b,c)`, it means, `2` divides `b` and `c` too and therefore `b` and `c` are even.

Hence as `a` and `c` both are even, they have a common factor `2` and hence `2` is a factor of `gcf(a,c)` too and divides `gcf(a,c)`.