Thread: Divisibility...

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)

Originally Posted by Vedicmaths

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

Originally Posted by Vedicmaths

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)

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.