A Pattern Emerges

Number Theory Level 2

Let x , y , z x, y, z be distinct positive integers . x x and y y are odd, and z z is even. Which one of the following statements cannot be true?

( x − z ) 2 y (x-z)^{2}y is even ( x + y ) 3 z (x+y)^{3}z is even ( x − y ) 2 z (x-y)^{2}z is even ( x − z ) y (x-z)y is odd ( x − z ) y 2 (x-z)y^{2} is odd

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Armain Labeeb by Armain Labeeb , 20, Bangladesh

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1 solution

Armain Labeeb
Jul 11, 2016

Relevant wiki: Even and Odd Numbers

Basics:

Even ± Even = Even Even ± Odd = Odd Odd ± Odd = Even Even × Even = Even Even × Odd = Even Odd × Odd = Odd \large{\text{Even ± Even = Even}\\ \text{Even ± Odd = Odd}\\ \text{Odd ± Odd = Even}\\ \text{Even × Even = Even}\\ \text{Even × Odd = Even}\\ \text{Odd × Odd = Odd} }

Apply these formulae to all the options and you will find ( x − z ) 2 y is even (x-z)^{2}y \text{ is even} is the only untrue option.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

@Armain Labeeb In general, we avoid having titles that say "easy" or "hard", because that is very subjective to a person. As such, we have removed your original title.

Calvin Lin Staff - 4 years, 11 months ago

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Oh alright. No problem :)

Armain Labeeb - 4 years, 11 months ago

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