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x and y are positive integers. If xy + x is odd, then which of the fol
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06 Jul 2021, 08:25
06 Jul 2021, 08:25
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x and y are positive integers. If xy + x is odd, then which of the following must be even?
A. x
B. y
C. x + y
D. xy − x
E. x^2 − y
A. x
B. y
C. x + y
D. xy − x
E. x^2 − y
Re: x and y are positive integers. If xy + x is odd, then which of the fol
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06 Jul 2021, 09:02
06 Jul 2021, 09:02
Solution:
We know that xy+x= odd
This is possible when odd+even=odd or even+odd=odd
Considering the first case, O+E=O means that x is even. If x is even, then xy is also even as any number multipled to an even integer is also even. We can rule out this option.
Considering the second case, E+O=O means x is odd and xy is even. Thus, y is even to get an even product when multipled by an odd integer.
IMO B
Hope this helps!
We know that xy+x= odd
This is possible when odd+even=odd or even+odd=odd
Considering the first case, O+E=O means that x is even. If x is even, then xy is also even as any number multipled to an even integer is also even. We can rule out this option.
Considering the second case, E+O=O means x is odd and xy is even. Thus, y is even to get an even product when multipled by an odd integer.
IMO B
Hope this helps!
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Re: x and y are positive integers. If xy + x is odd, then which of the fol
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06 Jul 2021, 09:10
06 Jul 2021, 09:10
1
Carcass wrote:
x and y are positive integers. If xy + x is odd, then which of the following must be even?
A. x
B. y
C. x + y
D. xy − x
E. x^2 − y
A. x
B. y
C. x + y
D. xy − x
E. x^2 − y
Some important rules regarding the products of evens and odds:
1. (ODD)(ODD) = ODD
2. (ODD)(EVEN) = EVEN
3. (EVEN)(EVEN) = EVEN
4. ODD +/- ODD = EVEN
5. ODD +/- EVEN = ODD
6. EVEN +/- EVEN = EVEN
As we can see from #1, IF the product of two integers is ODD, then the two integers must both be odd.
Given: xy + x is ODD
If we FACTOR xy + x, we get x(y + 1)
So, we're told that x(y + 1) is ODD
From rule #1 above, we can conclude that x is ODD and (y + 1) is ODD
If (y + 1) is odd, then we can be certain that y is EVEN
Now check the answer choices:
A. x = ODD
B. y = EVEN (done!!)
C. x + y = ODD + EVEN = ODD
D. xy − x = (ODD)(EVEN) - ODD = EVEN - ODD = ODD
E. x^2 − y = (ODD)(ODD) - EVEN = ODD - EVEN = ODD
Answer: B
