Proof by Contrapositive?

Question

Prove by contrapositive:

Let a,b,c,x,y be integers if a|b and a|c then a|bx+cy.

So I want to prove that if a doesn't divide bx+cy then a doesn't divide b and a doesnt divide c. But I don't really know how to get started.

Okay, so it should actually be:
If a doesn't divide bx+cy then a doesn't divide be OR a doesn't divide c.

Answer 1

Here is a possible proof:

Suppose a doesn't divide bx + cy.
Then (bx+cy)/a is not an integer.
In other words, (bx/a) + (cy/a) is not an integer (*).

I will now show that that a doesn't divide b OR c assuming the opposite and arriving at a contradiction.

Suppose a|b and a|c.
Then bx/a is an integer AND cy/a is an integer.
But this forces bx/a + cy/a to be an integer.
This contradicts (*).

Answer 2

Assume 'a' doesn't divide the sum & examine what that means for each term.

'a' doesn't divide 'bx'. So, 'a' doesn't divide 'b' or 'x'.
Similarly, 'a' doesn't divide 'cy', which implies, 'a' doesn't divide 'c' or 'y'.

Answer 3

The contrapositve is tricker for statements with and/or in thme.

What you have is

(A and B) implys C

The contrapositive of this is:

not C implys (not A or not B)

Answer 4

the contrapositive of every statment is true if the statement is true
look at the truth tables for both, they match
a [] b [] a->b [][] a [] b [] ~a [] ~b [] ~b->~a
T [] T [] T [][] T [] T [] F [] F [] T
F [] T [] F [][] F [] T [] T [] F [] F
T [] F [] T [][] T [] F [] F [] T [] T
F [] F [] T [][] F [] F [] T [] T [] T

Hope that helps =D
P.S.: i did the spacing cause the format didnt work with out it