If x and y are positive integer, is y an even integer?

Question

(1) x(x + 2) + 1 = xy

(2) x(x + y) is an odd integer

Archit3110 · Answer

target is y an even integer
#1
x(x + 2) + 1 = xy
 if x is odd then
LHS is even so at RHS y has to be even
case 2 : if x is even then
LHS is odd which is not possible at RHS as x is even
sufficient to say that y is even with x as odd
#2
 x(x + y) is an odd integer
this condition is possible with condition that x is odd and y is even
sufficient
option D is correct

argerniki · Answer

I think there are two approaches:

1. Plug in some numbers :

X, Y have to be integers, so I check what happens to Y when X is 1, 2, 3, 4

Statement (1) 
x(x + 2) + 1 = xy

x = 1 : 1(1 + 2) + 1 = 3+1 = 4 = 1*y --> y = 4 and even
 x = 2 : 2(2 + 2) + 1 = 8+1 = 9 = 2*y --> no int. solution for y because 9 is not div by 2
 x = 3 : 3(3 + 2) + 1 = 15+1 = 16 = 3*y --> no int. solution for y because 16 is not div by 3
 x = 4 : 4(4 + 2) + 1 = 24+1 = 25 = 4*y --> no int. solution for y because 25 is not div by 4

I deduce from this, that the only int. solution for y = 4, which is even

SUFF
(I'm not so secure about this solution, because I usually exlude 1 when I check numbers, so it took me a while)

Statement (2) 
x(x + y) is an odd integer

x = 1 : 1(1+y) = odd
If y = 1 (odd), then the equation is false
If y = 2 (even), then the equation is true

x = 2 : 2(2+y) = odd
Don't need to check further, because 2 * anything can't be odd

x = 3 : 3(3+y) = odd
If y = 1 (odd), then the equation is false
If y = 2 (even), then the equation is true

Therefore, in order to be true, x has to be odd and y has to be even
SUFF
Answer D

1. Number Theory :

Try to translate the terms into even & odd:

Statement (1) 
x(x + 2) + 1 = xy

Case 1: x is even
(x+2) is even -> e * e + 1 = xy
Since e * e = even, (e * e)+1 = odd
Therefore: xy = odd
Since x = even, this case can't be true

Case 2: x is odd
(x+2) is odd -> o * o + 1 = xy
Since o * o = odd, (o * o)+1 = even
Therefore: xy = even
Since x = odd in this case, y has to be even

Therefore, x has to be odd and y has to be even
SUFF

Statement (2) 
x(x + y) is an odd integer

Case 1: x is even
e * (e+y) = odd
Since one factor is even, the whole number can't be odd

Case 2: x is odd
o * (o+y) = odd
When y = even --> o+e = o --> o * o = o
When y = odd --> o+o = e --> can't be true, because one factor is even

Therefore, x has to be odd and y has to be even
SUFF
Answer D

If x and y are positive integer, is y an even integer?

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If x and y are positive integer, is y an even integer?

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