SOLUTION: If 5x-y is divisible by 4 and 2x+3y is odd. Then 7x+2y is odd for all x,y ∈Z. Prove by direct proof or contrapositive

Question 1194180: If 5x-y is divisible by 4 and 2x+3y is odd. Then 7x+2y is odd for all x,y ∈Z. Prove by direct proof or contrapositive
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
.
If 5x-y is divisible by 4 and 2x+3y is odd. Then 7x+2y is odd for all x,y ∈Z. Prove by direct proof or contrapositive
~~~~~~~~~~~~~

            To be mathematically correct (and do not scare readers), the statement needs to be modified.
            The modified version is THIS :

If x and y are integer numbers such that 5x-y is divisible by 4 and 2x+3y is odd, then 7x+2y is odd.

            Below is my solution for this modified version.

Since 2x + 3y is odd, it means that 3y is odd. In turn, it means that y is odd. Next, 5x-y is divisible by 4. It implies that x-y is divisible by 4. Since y is odd (see above), it means that x is odd. Then 7x is odd, and hence 7x + 2y is odd.

Proved and solved.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

If 5x-y is divisible by 4, then it's also divisible by 2 since 4 = 2*2

Put another way:
4 is a factor of 5x-y
5x-y = 4k for some integer k
5x-y = 2*(2k)
This shows that 5x-y is even

5x-y is even while 2x+3y is odd
(5x-y)+(2x+3y) = (5x+2x) + (-y+3y) = 7x+2y

We've summed an even number (5x-y) with an odd number (2x+3y) to get an odd result (7x+2y). This concludes the proof.

-----------------------------------------------

Here's a side proof showing that odd+even = odd
This is something for another page (perhaps a reference page in your notebook somewhere).

k = some integer
2k = some even integer
2m+1 = some odd integer

odd+even = (2m+1)+2k = (2m+2k)+1 = 2(m+k)+1 = some other odd integer

This concludes the proof that odd+even = odd
Since order doesn't matter when it comes to adding, it's the same as saying even+odd = odd

Some numeric examples to partially verify the claim:
3+2 = 5
7+14 = 21
19+18 = 37
These are examples to help see why the proof works. Of course you'd need the proof above to fully confirm the claim.

RELATED QUESTIONS
If 5x-y is divisible by 4 and 2x+3y is odd. Then 7x+2y is odd for all x,y... (answered by ikleyn)
x and y are integers. show that x^2-y^2 is odd or divisible by... (answered by Jk22)
Prove that if �a� and �b� are odd positive integers, then a2+b2 is even but not divisible (answered by richard1234)
If x,y,z is odd integer then the result of this is odd integer by using quadratic... (answered by ikleyn)
Prove that the product of two odd numbers is odd, using an indirect proof and a proof by... (answered by josgarithmetic,ikleyn)
proof by contradiction if 3n+1 is odd then n is also... (answered by fcabanski)
Show that if x^2 is odd, x is odd. Use proof by... (answered by ikleyn)
Prove: if 4 divides mn, then m is even or n is even. i saw that this was p implies q... (answered by richard1234)
what is the contrapositive of "If 3<4 then 3 is an odd... (answered by richwmiller)