Even / Odd Numbers - GRE Quantitative Reasoning
Card 1 of 312
If 
and 
are both odd integers, which of the following is not necessarily odd?
If
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With many questions like this, it might be easier to plug in numbers rather than dealing with theoretical variables. However, given that this question asks for the expression that is not always even or odd but only not necessarily odd, the theoretical route might be our only choice.
Therefore, our best approach is to simply analyze each answer choice.
: Since 
is odd, 
is also odd, since and odd number multiplied by an odd number yields an odd product. Since 
is also odd, multiplying it by 
will again yield an odd product, so this expression is always odd.
: Since 
is odd, multiplying it by 2 will yield an even number. Subtracting this number from 
will also give an odd result, since an odd number minus an even number gives an odd number. Therefore, this answer is also always odd.
: Since both numbers are odd, their product will also always be odd.
: Since 
is odd, multiplying it by 2 will give an even number. Since 
is odd, subtracting it from our even number will give an odd number, since an even number minus and odd number is always odd. Therefore, this answer will always be odd.
: Since both numbers are odd, there sum will be even. However, dividing an even number by another even number (2 in our case) does not always produce an even or an odd number. For example, 5 and 7 are both odd. Their sum, 12, is even. Dividing by 2 gives 6, an even number. However, 5 and 9 are also both odd. Their sum, 14, is even, but dividing by 2 gives 7, an odd number. Therefore, this expression isn't necessarily always odd or always even, and is therefore our answer.
With many questions like this, it might be easier to plug in numbers rather than dealing with theoretical variables. However, given that this question asks for the expression that is not always even or odd but only not necessarily odd, the theoretical route might be our only choice.
Therefore, our best approach is to simply analyze each answer choice.
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Theodore has 
jelly beans. Portia has three times that amount. Harvey has five times as many as she does. What is the total count of jelly beans in the whole group?
Theodore has
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To find the answer to this question, calculate the total jelly beans for each person:
Portia: 
* <Theodore's count of jelly beans>, which is 
or 
Harvey: 
* <Portia's count of jelly beans>, which is 
or 
So, the total is:
(Do not forget that you need those original 
for Theodore!)
To find the answer to this question, calculate the total jelly beans for each person:
Portia:
Harvey:
So, the total is:
(Do not forget that you need those original
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Assume 
and 
are both odd whole numbers.
What is a possible solution for 
?
Assume
What is a possible solution for
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When two odd whole numbers are multipliedd, they will equal an odd whole number. The only answer that fits the requirements of being an odd and a whole number is 
since it comes from 
or 
.
When two odd whole numbers are multipliedd, they will equal an odd whole number. The only answer that fits the requirements of being an odd and a whole number is
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What are three consecutive odd integers whose sum equals 
?
What are three consecutive odd integers whose sum equals
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Set up the equation,
.
Simplify to 
and solve for x to find 
.
Therefore the 3 consecutive odd integrers are 
.
Set up the equation,
Simplify to
Therefore the 3 consecutive odd integrers are
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Assume 
and 
are both odd whole numbers and 
.
What is a possible solution for 
?
Assume
What is a possible solution for
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The two requirements for this problem are that both 
and 
must be odd, and that 
. The only answer that fits both of these is 
. The other answers show either 
or 
is an even number.
The two requirements for this problem are that both
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Solve for 
:
Solve for
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To solve, divide both sides of the equation by 
:
As a check, if you divide an odd number by another odd number, your result should be odd.
To solve, divide both sides of the equation by
As a check, if you divide an odd number by another odd number, your result should be odd.
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Solve for 
:
Solve for
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To solve, isolate your variable by dividing both sides of the equation by 
:
To solve, isolate your variable by dividing both sides of the equation by
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Solve for 
:
Solve for
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To solve, first isolate your variable by dividing both sides of the equation by 
:
As a check, if you divide an odd number by another odd number, your result should be odd.
To solve, first isolate your variable by dividing both sides of the equation by
As a check, if you divide an odd number by another odd number, your result should be odd.
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Solve for 
:
Solve for
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To solve, divide both sides of the equation by 
:
As a check, if you divide an odd number by another odd number, your result should be odd.
To solve, divide both sides of the equation by
As a check, if you divide an odd number by another odd number, your result should be odd.
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Assume 
and 
are both even whole numbers and 
.
What is a possible solution for 
?
Assume
What is a possible solution for
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Since 
, 
must result in a positive whole number. The only answer that fits these requirements of being both positive and whole number is 
.
Since
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Choose the answer below which best solves the following equation:
Choose the answer below which best solves the following equation:
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If it's simpler for you, you can split this problem into two parts: First, take 
away from 
, and you're left with 
. Then, take away 
from 
(you can even count backwards if necessary), and you'll be left with the final answer, 
.
If it's simpler for you, you can split this problem into two parts: First, take
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A bus has sixteen passengers at its first stop. It drops off three at the second stop, and three at the third stop. At the fourth stop, eveyone else gets off the bus. How many people got off at the fourth stop?
A bus has sixteen passengers at its first stop. It drops off three at the second stop, and three at the third stop. At the fourth stop, eveyone else gets off the bus. How many people got off at the fourth stop?
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First, add the total number of passengers that got off BEFORE the fourth stop. Three plus three is six, so you know that you've lost six total passengers before the fourth stop. 
, so there are ten passengers remaining at the fourth stop, and that's how many get off there.
First, add the total number of passengers that got off BEFORE the fourth stop. Three plus three is six, so you know that you've lost six total passengers before the fourth stop.
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The sum of seven consecutive even integers is 0.
Column A: The product of the seven integers
Column B: 2
The sum of seven consecutive even integers is 0.
Column A: The product of the seven integers
Column B: 2
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For the sum of 7 consecutive even integers to be zero, the only sequence possible is –6, –4, –2, 0, 2, 4, 6. This can be determined algebraically by assigning the lowest number in the sequence to be “y” and adding 2 for each consecutive even integer, and then setting this equal to zero.
y, y + 2, y + 4, y + 6 . . .
The product of any number and 0 is 0.
Therefore, column B must be greater.
For the sum of 7 consecutive even integers to be zero, the only sequence possible is –6, –4, –2, 0, 2, 4, 6. This can be determined algebraically by assigning the lowest number in the sequence to be “y” and adding 2 for each consecutive even integer, and then setting this equal to zero.
y, y + 2, y + 4, y + 6 . . .
The product of any number and 0 is 0.
Therefore, column B must be greater.
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and 
are both even whole numbers.
What is a possible solution for 
?
What is a possible solution for
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If 
and 
are both even whole numbers, then their addition must be an even whole number as well. Although 
is an even number, it is not a whole number and could therefore not be a solution. This means the only possible solution would be 
.
If
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If John buys twenty-two apples on Monday, and thirty-four bananas on Tuesday, how many pieces of fruit does John have?
If John buys twenty-two apples on Monday, and thirty-four bananas on Tuesday, how many pieces of fruit does John have?
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Plugging in the values given we arrive at the total fruit John has:
A good note about adding even numbers--any even numbers--is that if you add two even numbers, their sum will ALWAYS be an even number.
Plugging in the values given we arrive at the total fruit John has:
A good note about adding even numbers--any even numbers--is that if you add two even numbers, their sum will ALWAYS be an even number.
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Choose the answer below which best solves the following problem:
Choose the answer below which best solves the following problem:
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To deal with a problem with this many digits, often the best strategy is to line up one number over the other, then add the places one at a time. Don't forget to carry a one every time the addition goes over ten. Also, note that any time you add two even numbers, their sum will ALWAYS be an even number.
To deal with a problem with this many digits, often the best strategy is to line up one number over the other, then add the places one at a time. Don't forget to carry a one every time the addition goes over ten. Also, note that any time you add two even numbers, their sum will ALWAYS be an even number.
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If 
and 
are odd integers, and 
is even, which of the following must be an odd integer?
If
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Even numbers come in the form 2x, and odd numbers come in the form (2x + 1), where x is an integer. If this is confusing for you, simply plug in numbers such as 1, 2, 3, and 4 to find that:
Any odd number + any even number = odd number
Any odd number + any odd number = even number
Any even number x any number = even number
Any odd number x any odd number = odd number
a(b + c) = odd x (odd + even) = odd x (odd) = odd
Even numbers come in the form 2x, and odd numbers come in the form (2x + 1), where x is an integer. If this is confusing for you, simply plug in numbers such as 1, 2, 3, and 4 to find that:
Any odd number + any even number = odd number
Any odd number + any odd number = even number
Any even number x any number = even number
Any odd number x any odd number = odd number
a(b + c) = odd x (odd + even) = odd x (odd) = odd
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If m, n and p are odd integers, which of the following must be an odd integer?
If m, n and p are odd integers, which of the following must be an odd integer?
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When multiplying odd/even numbers, we know that odd * odd = odd, and odd * even = even. We also know that odd + odd = even. We will proceed to evaluate each answer choice, knowing that m, n and p are odd.
(m + 1) * n
m + 1 becomes even. This gives us even * odd = even.
m + n + p + 1
Odd + odd + odd + odd = even + odd + odd = even + even = even.
(m - 2 )* n * p
m - 2 is stil odd. This gives us odd * odd * odd = odd * odd = odd.
m * (n + p)
Odd + odd is even, so here we have odd * even = even.
m * p * (n -1)
n - 1 becomes even so we have odd * odd * even = odd * even = even.
The correct answer is therefore m * p * (n -1).
When multiplying odd/even numbers, we know that odd * odd = odd, and odd * even = even. We also know that odd + odd = even. We will proceed to evaluate each answer choice, knowing that m, n and p are odd.
(m + 1) * n
m + 1 becomes even. This gives us even * odd = even.
m + n + p + 1
Odd + odd + odd + odd = even + odd + odd = even + even = even.
(m - 2 )* n * p
m - 2 is stil odd. This gives us odd * odd * odd = odd * odd = odd.
m * (n + p)
Odd + odd is even, so here we have odd * even = even.
m * p * (n -1)
n - 1 becomes even so we have odd * odd * even = odd * even = even.
The correct answer is therefore m * p * (n -1).
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The operation ¤ is defined for all integers x and y as: x ¤ y = 4x-y2.
If x and y are positive integers, which of the following cannot produce an odd value?
The operation ¤ is defined for all integers x and y as: x ¤ y = 4x-y2.
If x and y are positive integers, which of the following cannot produce an odd value?
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For this problem, we must recognize under what arithmetic conditions an even or odd number is produced. We do not know what the values for x and y are, but we do know for example that any two numbers added can be either even or odd, and and any number multiplied by two must be even (e.g. 2 * 2 = 4; 3 * 2 = 6), or an odd number multiplied by another odd is always even.
In this situation, we have to ensure that an arithmetic operation (subtraction) must result only in an even number, by further ensuring that the variables themselves are fixed to either odd or even values.
In this situation, we know that due to the "*2 (= 2 * 2)" principle, the 4x component always being an even value. The only situation in which the operation "4x – y2" will be even is if the "y2" term is also even, since an even minus an odd would be odd. The only situation in which the "y2" term is even is if the y itself is even, since an odd number squared always results in an odd value (e.g. 32 = 9; 72 = 25). To ensure that the y itself is even, we must also double it to "2y"(2).
We cannot add 1 to a random variable y as it may still result in an odd y-value. Similarly, since we already stated that a squared value may still be odd, we cannot be sure that squaring the y will also result in an even number.
For this problem, we must recognize under what arithmetic conditions an even or odd number is produced. We do not know what the values for x and y are, but we do know for example that any two numbers added can be either even or odd, and and any number multiplied by two must be even (e.g. 2 * 2 = 4; 3 * 2 = 6), or an odd number multiplied by another odd is always even.
In this situation, we have to ensure that an arithmetic operation (subtraction) must result only in an even number, by further ensuring that the variables themselves are fixed to either odd or even values.
In this situation, we know that due to the "*2 (= 2 * 2)" principle, the 4x component always being an even value. The only situation in which the operation "4x – y2" will be even is if the "y2" term is also even, since an even minus an odd would be odd. The only situation in which the "y2" term is even is if the y itself is even, since an odd number squared always results in an odd value (e.g. 32 = 9; 72 = 25). To ensure that the y itself is even, we must also double it to "2y"(2).
We cannot add 1 to a random variable y as it may still result in an odd y-value. Similarly, since we already stated that a squared value may still be odd, we cannot be sure that squaring the y will also result in an even number.
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What is the 65th odd number?
What is the 65th odd number?
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Doing this by counting the odd numbers will take way too long for the GRE. And if you look at the answer choices, you see five consecutive odd numbers, so one little mistake in counting will give you the wrong answer! Instead, we should use the formula for finding odd numbers: the n_th odd number is 2_n – 1. (The n_th even number is 2_n.) So the 65th odd number is 2 * 65 – 1 = 129.
Doing this by counting the odd numbers will take way too long for the GRE. And if you look at the answer choices, you see five consecutive odd numbers, so one little mistake in counting will give you the wrong answer! Instead, we should use the formula for finding odd numbers: the n_th odd number is 2_n – 1. (The n_th even number is 2_n.) So the 65th odd number is 2 * 65 – 1 = 129.
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