Sunday, July 14, 2013

# Ode to Odd and Even Integers! GRE Math fact

2:33 AM
While the positive or negative classification can apply to integers and decimals/ fractions, odd or even is a classification that only applies to integers.
Even integers are integers that are divisible by 2 i.e. when an even number is divided by 2, there is a zero remainder. All even integers must end in 0, 2, 4, 6, or 8 i.e. their unit’s digit must 0, 2, 4, 6 or 8. An even integer can be represented as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 2n$where $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$is any integer. The following table shows values of even number for different values of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$

 $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$ Even Number ( $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 2n$) 0 0 1 2 2 4 3 6
Odd integers are those that are not divisible by 2 i.e. when an odd number is divided by 2, there is always a remainder that is equal to 1. All odd integers must end in 1, 3, 5, 7, or 9 i.e. their unit’s digit must 1, 3, 5, 7, or 9. An odd integer can be represented as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 2n+1$where $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$is any integer. The following table shows values of even number for different values of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$
 $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$ Odd Number ( $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} 2n+1$) 0 1 1 3 2 5 3 7

There are certain rules involving operations such as addition, subtraction, multiplication or division. Remembering them may be helpful to solve certain types of problems:
Addition and Subtraction Rules for Odd and Even Integers
The sum of any number of even integers is always even.
The sum of even number of odd integers is even, the sum of odd number of odd integers is odd.
The difference of two even integers is always even. Even 0 is an even integer for that matter.
The difference of two odd integers is always even.
The following Table summarizes these rules for addition and subtraction:
 Even Odd Even Even Odd Odd Odd Even
The following table summarizes these rules for multiplication
 Even Odd Even Even Even Odd Even Odd
When dividing two integers, it is impossible to determine whether the result will be an integer or a fraction/decimal.  Therefore there are no hard and fast rules for division of integers.
Let us look at some examples where the above concepts are used:
Example:
If $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} x$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} y$ are odd numbers, which of the following can never be an even number?
A) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} x+y$
B) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 3x+y$
C) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 2x+4y$
D) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 3x+2y$
E) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 4xy$
Solution:

This problem has to be solved using the process of elimination. Read the question carefully, it wants you to identify the expression that can never be an even number. Therefore any expression that is reduced to an even number can be eliminated.
Answer choice A) is incorrect. Since $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} x$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} y$ are odd numbers, we know that $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} x+y$ must be even since sum of two odd numbers is always even
Answer choice B) is incorrect. $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 3x+y$ must be even since $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 3x$ is odd as well as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} y$ are odd number. Since sum of two odd numbers is always even, answer choice B) is incorrect.
Answer choice C) is obviously incorrect since $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 2x+4y=2(x+2y)$ must be even.
Answer choice D) is correct. Since $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} x$ is odd, $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 3x$ must be odd. In addition, regardless of the value of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} y$ ,$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 2y$ must be even. Since the sum of odd and even number must be odd, answer choice D) is correct
Answer choice E) is also incorrect, since $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 4xy$ must be a multiple of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi{120} 4$ and hence must be an even number.