Conjectures about Integers

Conjectures about Integers test if you are familiar with properties of integers and their classifications. Review Types of Integers if you do not know the definition of even, odd, positive, negative, consecutive integers, prime numbers, etc.

If xx and yy are both integers, which of the following must be even?

A) x+y x + y .
B) 2x+y 2x + y .
C) x2+y2 x^2 + y^2 .
D) (x+y)2(xy)2 (x+y)^2 - ( x - y)^ 2 .
E) x2+xy+y2 x^2 + xy + y^2 .

Solution: Let's consider A. If x=1,y=2 x = 1, y = 2 then x+y=3 x + y = 3 which is not even.
Let's consider B. If x=2,y=1 x = 2, y =1 , then 2x+y=5 2x+y =5 which is not even.
Let's consider C. If x=1,y=2 x = 1, y =2 then x2+y2=5 x^2 + y^2 = 5 which is not even.
Let's consider D. (x+y)2(xy)2=(x2+2xy+y2)(x22xy+y2)=4xy (x+y)^2 - (x-y)^2 = ( x^2 + 2xy + y^2) - ( x^2 - 2xy + y^2) = 4xy . Since we're multiplying by 4, hence this must be even.
Let's consider E. If x=1,y=2 x = 1, y = 2 , then x2+xy+y2=1+2+4=7 x^2 + xy + y^2 = 1 + 2 + 4 = 7 which is not even.
Hence the answer is D.

Note by Arron Kau
5 years, 2 months ago

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