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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(52798)   (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
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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.
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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.
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