OG (12th ed) DS Prob #106

[meanjonathan]

Fairly straightforward problem...

If x and y are integers, is xy even?

1) x=y+1
2) x/y is an even integer.

Answer guide says D.

How does statement 1) eliminate the possibility of x or y being 0? For instance, y=0, x=0+1, 0*1=0; non-even... or y=-1, x=0, -1*0=0

What am I missing?

[Anurag@Gurome]

Fairly straightforward problem...

If x and y are integers, is xy even?

1) x=y+1
2) x/y is an even integer.

Answer guide says D.

How does statement 1) eliminate the possibility of x or y being 0? For instance, y=0, x=0+1, 0*1=0; non-even... or y=-1, x=0, -1*0=0

What am I missing?

Note that 0 is an even integer.

Question: Is xy an even integer?

(1) x = y + 1
If y = 0, then x = 1 and xy = 0, which is an even integer.
If y = 1, then x = 2 and xy = (2)(1) = 2, which is an even integer.
If y = 2, then x = 3 and xy = (2)(3) = 6, which is an even integer.
In general, if y = even, x = even + 1 = odd, and xy = odd * even = even
If y = odd, x = odd + 1 = even and xy = even * odd = even; SUFFICIENT.

(2) x/y is an even integer implies both x and y are even integers. Hence xy will also be an even integer; SUFFICIENT.

The correct answer is D.

Hope that helps.

[Shalabh's Quants]

Fairly straightforward problem...

If x and y are integers, is xy even?

1) x=y+1
2) x/y is an even integer.

Answer guide says D.

How does statement 1) eliminate the possibility of x or y being 0? For instance, y=0, x=0+1, 0*1=0; non-even... or y=-1, x=0, -1*0=0

What am I missing?

Stat. 1...

x=y+1

=> x=y+odd;

If y is odd then x=odd+odd=even, then x.y=odd.even=even

If y is even then x=even+odd=odd,then x.y=even.odd=even

Stat. 1 is suff.

Stat. 2...

x/y is an even integer.

For Division (If completely divisible) of 2 integers to be even,

(1) Either Numerator should be Even and Denominator should be Odd. Even/Odd=Even (6/3=2; 100/25=4 etc.). It means x is Even so XY will be even.

(2) Or If both Numerator & Denominator are Even, then result may be Even or Odd [4/2=2(Even); 10/2=5(odd)]. In our case x/y is even(given), It means x & Y both are Even so XY will be even.

Stat. 2 is suff. Ans. D.

[Birottam Dutta]

The two important points to note in this problem are :

1. 0 is even
2. Odd * even = Even, Even * Even = Even, odd* odd= Odd.

In statement 1, X= Y+1. So, if y is even, x is odd and vice versa. In either case, xy will be even.

In statement 2, x/y is an even integer. Let, x/y = k, where k is the even integer, then x= k*y = even irrespective of whether y is even or odd. Hence, in this case also, x*y will be even as x is even.

You only need one even term to make the product of two numbers even.

[meanjonathan]

0 is even. That resolves the descrepancy. Thanks.