Show that if integers a and b both are not divisible by 5, then the product ab is also not divisible by 5....where a=5k+r such as (1 <= r <= 4)
and . Since 5 does not divide, .
Now we want to prove that 5 does not divide ab. Lets do what Jhevon has taught you before.... the contradiction method....
So let us assume 5 does divide ab, that means 5 divides . But this means 5 divides . But since and are both less than 5 and and nonzero, 5 cannot divide . This contradicts our assumption that 5 divides ab. So 5 does not divide ab
Another way to do it.
Lets write this as an if then statement
if and
then
in this form we can find p and q
is logically equivelent to (the contraposition)
or in words
if then or
this is easier to prove
since 5|ab
suppose that 5|a then we are done...
now suppose that we need to show that 5|b
but 5=(ab)q 5=(aq)b so 5|b
QED
I hope this helps.