Even / Odd Numbers - PSAT Math
Card 1 of 231
If x and y are integers and at least one of them is even, which of the following MUST be true?
If x and y are integers and at least one of them is even, which of the following MUST be true?
Tap to reveal answer
Since we are only told that "at least" one of the numbers is even, we could have one even and one odd integer OR we could have two even integers.
Even plus odd is odd, but even plus even is even, so x + y could be either even or odd.
Even times odd is even, and even times even is even, so xy must be even.
Since we are only told that "at least" one of the numbers is even, we could have one even and one odd integer OR we could have two even integers.
Even plus odd is odd, but even plus even is even, so x + y could be either even or odd.
Even times odd is even, and even times even is even, so xy must be even.
← Didn't Know|Knew It →
If n and m are both positive even integers, which of the following must be odd?
I. (n + 1)(m + 1)
II. nm + 1
III. nm + m
If n and m are both positive even integers, which of the following must be odd?
I. (n + 1)(m + 1)
II. nm + 1
III. nm + m
Tap to reveal answer
Let us analyze I, II, and III one at a time.
Because n and m are both even, if we increase either by 1, the result will be an odd number. Thus, n + 1 and m + 1 are both odd. When two odd numbers are multiplied together, the result is always an odd number. Thus (n + 1)(m + 1) must be an odd number.
Because n and m are even, when we multiply two even numbers together, we always get an even number. Thus nm is even. However, when we then add one to an even number, the result will be an odd number. Thus, nm + 1 is odd.
We just established that nm is even. If we subtract an even number from an even number, the result is always even. Thus, nm – m is an even number.
Only choice I and II will always produce odd numbers.
The answer is I and II only.
Let us analyze I, II, and III one at a time.
Because n and m are both even, if we increase either by 1, the result will be an odd number. Thus, n + 1 and m + 1 are both odd. When two odd numbers are multiplied together, the result is always an odd number. Thus (n + 1)(m + 1) must be an odd number.
Because n and m are even, when we multiply two even numbers together, we always get an even number. Thus nm is even. However, when we then add one to an even number, the result will be an odd number. Thus, nm + 1 is odd.
We just established that nm is even. If we subtract an even number from an even number, the result is always even. Thus, nm – m is an even number.
Only choice I and II will always produce odd numbers.
The answer is I and II only.
← Didn't Know|Knew It →
odd * odd * odd =
odd * odd * odd =
Tap to reveal answer
The even/odd number properties are good to know. If you forget them, however, it's easy to check with an example.
Odd * odd = odd. If you didn't remember that, a check such as 1 * 3 = 3 will give you the same answer. So if odd * odd = odd, (odd * odd) * odd = odd * odd = odd, just as 3 * 3 * 3 = 27, which is odd. This means we are looking for an answer choice that also produces an odd number. Let's go through them.
even * even = even (2 * 2 = 4)
even * odd = even (2 * 3 = 6)
odd * odd = odd (1 * 3 = 3) This is the correct answer! But just to double check, let's go through the last two.
even * even * even = even * even = even (2 * 2 * 2 = 8)
odd * odd * even = odd * even = even (1 * 3 * 2 = 6)
The even/odd number properties are good to know. If you forget them, however, it's easy to check with an example.
Odd * odd = odd. If you didn't remember that, a check such as 1 * 3 = 3 will give you the same answer. So if odd * odd = odd, (odd * odd) * odd = odd * odd = odd, just as 3 * 3 * 3 = 27, which is odd. This means we are looking for an answer choice that also produces an odd number. Let's go through them.
even * even = even (2 * 2 = 4)
even * odd = even (2 * 3 = 6)
odd * odd = odd (1 * 3 = 3) This is the correct answer! But just to double check, let's go through the last two.
even * even * even = even * even = even (2 * 2 * 2 = 8)
odd * odd * even = odd * even = even (1 * 3 * 2 = 6)
← Didn't Know|Knew It →
When 
is evaluated, what number is in the ones digit?
When
Tap to reveal answer
Let us say we have two numbers, 
and 
whose ones digits are 
and B, respectively. If we want to know the ones digits of the product of 
and 
, all we need to do is to look at the ones digit of the product of 
and 
. For example, if we multiply 137 and 219, then the ones digit will be the same as the ones digit of 
. Since the ones digit of 63 is 3, the ones digit of 137 x 219 will also be 3. In short, we really only need to worry about the ones digits of the numbers we multiply when we try to find the ones digit of their product.
We want to find the ones digit of 
. An exponent is essentially just a short hand for repeated multiplication. Let us look at the ones digit of the first few exponents of 2013.
- the ones digit is 3.
To find the ones digit of the 2013 to the second power, we need to think of it as the product of 2013 and 2013. As discussed previously, if we want the ones digit of two numbers multiplied together, we just need to multiply their ones digits. Thus, if we multiply 2013 by 2013, then the ones digit will be the same as 
.
- ones digit is 9.
Next, we want to find the ones digit of 2013 to the third power. In order to do this, we will multiply the square of 2013 by 2013. It does not matter that we do not know exactly what 2013 squared equals, beacuse we only need to worry about the ones digit, which is 9. In other words, 2013 to the third power will have a ones digit that is equal to the ones digit of the product of 9 (which was the ones digit of 2013 squared) and 3 (which is the ones digit of 2013). When we multiply 9 and 3, we get 27, so the ones digit of 2013 to the third power is 7.
- ones digit of 7.
To find the ones digit of 2013 to the fourth power, we only need to worry about multiplying the ones digit of 2013 to the third power (which is 7) by the ones digit of 2013. When we mulitply 7 and 3, we get 21, which means that the ones digit of 2013 to the fourth power is 1.
- ones digit of 1.
To find the ones digit of 2013 to the fifth power, we will multiply 1 by 3, which gives us 3.
- ones digit of 3.
Notice that we are back to a ones digit with 3. If we multiply this by 2013, we will end up with a ones digit of 9. In other words, the ones digits repeat every fourth power.
The value of the ones digits of the powers of 2013 is as follows (starting with 2013 to the first power):
3, 9, 7, 1, 3, 9, 7, 1,....
We essentially want to find the 2013th term of the sequence above. Notice that every fourth term is 1, i.e. the sequence repeats every four terms. If a terms position in the sequence is a multiple of 4, then the term will be 1. In short, the 4th, 8th, 12th, 16th terms, and so on, will be 1. Because 2012 is a multiple of 4, the 2012th term in the sequence will be 1. (We can determine if a number is a multiple of 4 by looking at its last two digits.) This means that that 2013th term will be 3. Thus, 2013 to the power of 2013 has a ones digit of 3.
The answer is 3.
Let us say we have two numbers,
We want to find the ones digit of
To find the ones digit of the 2013 to the second power, we need to think of it as the product of 2013 and 2013. As discussed previously, if we want the ones digit of two numbers multiplied together, we just need to multiply their ones digits. Thus, if we multiply 2013 by 2013, then the ones digit will be the same as
Next, we want to find the ones digit of 2013 to the third power. In order to do this, we will multiply the square of 2013 by 2013. It does not matter that we do not know exactly what 2013 squared equals, beacuse we only need to worry about the ones digit, which is 9. In other words, 2013 to the third power will have a ones digit that is equal to the ones digit of the product of 9 (which was the ones digit of 2013 squared) and 3 (which is the ones digit of 2013). When we multiply 9 and 3, we get 27, so the ones digit of 2013 to the third power is 7.
To find the ones digit of 2013 to the fourth power, we only need to worry about multiplying the ones digit of 2013 to the third power (which is 7) by the ones digit of 2013. When we mulitply 7 and 3, we get 21, which means that the ones digit of 2013 to the fourth power is 1.
To find the ones digit of 2013 to the fifth power, we will multiply 1 by 3, which gives us 3.
Notice that we are back to a ones digit with 3. If we multiply this by 2013, we will end up with a ones digit of 9. In other words, the ones digits repeat every fourth power.
The value of the ones digits of the powers of 2013 is as follows (starting with 2013 to the first power):
3, 9, 7, 1, 3, 9, 7, 1,....
We essentially want to find the 2013th term of the sequence above. Notice that every fourth term is 1, i.e. the sequence repeats every four terms. If a terms position in the sequence is a multiple of 4, then the term will be 1. In short, the 4th, 8th, 12th, 16th terms, and so on, will be 1. Because 2012 is a multiple of 4, the 2012th term in the sequence will be 1. (We can determine if a number is a multiple of 4 by looking at its last two digits.) This means that that 2013th term will be 3. Thus, 2013 to the power of 2013 has a ones digit of 3.
The answer is 3.
← Didn't Know|Knew It →
odd * odd * odd =
odd * odd * odd =
Tap to reveal answer
The even/odd number properties are good to know. If you forget them, however, it's easy to check with an example.
Odd * odd = odd. If you didn't remember that, a check such as 1 * 3 = 3 will give you the same answer. So if odd * odd = odd, (odd * odd) * odd = odd * odd = odd, just as 3 * 3 * 3 = 27, which is odd. This means we are looking for an answer choice that also produces an odd number. Let's go through them.
even * even = even (2 * 2 = 4)
even * odd = even (2 * 3 = 6)
odd * odd = odd (1 * 3 = 3) This is the correct answer! But just to double check, let's go through the last two.
even * even * even = even * even = even (2 * 2 * 2 = 8)
odd * odd * even = odd * even = even (1 * 3 * 2 = 6)
The even/odd number properties are good to know. If you forget them, however, it's easy to check with an example.
Odd * odd = odd. If you didn't remember that, a check such as 1 * 3 = 3 will give you the same answer. So if odd * odd = odd, (odd * odd) * odd = odd * odd = odd, just as 3 * 3 * 3 = 27, which is odd. This means we are looking for an answer choice that also produces an odd number. Let's go through them.
even * even = even (2 * 2 = 4)
even * odd = even (2 * 3 = 6)
odd * odd = odd (1 * 3 = 3) This is the correct answer! But just to double check, let's go through the last two.
even * even * even = even * even = even (2 * 2 * 2 = 8)
odd * odd * even = odd * even = even (1 * 3 * 2 = 6)
← Didn't Know|Knew It →
If 
and 
are both odd integers, which of the following is not necessarily odd?
If
Tap to reveal 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.
: 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.
← Didn't Know|Knew It →
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
Tap to reveal answer
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
← Didn't Know|Knew It →
If 
is an odd integer and 
is an even integer, which of the following must true of 
?
If
Tap to reveal answer
An even number subtracted from an odd number will always produce an odd result.
None of the other answer choices are correct.
An even number subtracted from an odd number will always produce an odd result.
None of the other answer choices are correct.
← Didn't Know|Knew It →
Which of the following could represent the sum of 3 consecutive odd integers, given that d is one of the three?
Which of the following could represent the sum of 3 consecutive odd integers, given that d is one of the three?
Tap to reveal answer
If the largest of the three consecutive odd integers is d, then the three numbers are (in descending order):
d, d – 2, d – 4
This is true because consecutive odd integers always differ by two. Adding the three expressions together, we see that the sum is 3_d_ – 6.
If the largest of the three consecutive odd integers is d, then the three numbers are (in descending order):
d, d – 2, d – 4
This is true because consecutive odd integers always differ by two. Adding the three expressions together, we see that the sum is 3_d_ – 6.
← Didn't Know|Knew It →
If n is an integer that is not equal to 0, which of the following must be greater than or equal to n?
I. 7n
II. n + 5
III. n2
If n is an integer that is not equal to 0, which of the following must be greater than or equal to n?
I. 7n
II. n + 5
III. n2
Tap to reveal answer
I is not always true because a negative number multiplied by 7 will give a number that is more negative than the original. II is true because adding 5 to any number will increase the value. III is true because squaring any number will increase the magnitude of the value, and squaring a negative number will make it positive.
I is not always true because a negative number multiplied by 7 will give a number that is more negative than the original. II is true because adding 5 to any number will increase the value. III is true because squaring any number will increase the magnitude of the value, and squaring a negative number will make it positive.
← Didn't Know|Knew It →
If x is an even integer and y is an odd integer. Which of these expressions represents an odd integer?
I. xy
II. x-y
III. 3x+2y
If x is an even integer and y is an odd integer. Which of these expressions represents an odd integer?
I. xy
II. x-y
III. 3x+2y
Tap to reveal answer
I)xy is Even*Odd is Even. II) x-y is Even+/-Odd is Odd. III) 3x is Odd*Even =Even, 2y is Even*Odd=Even, Even + Even = Even. Therefore only II is Odd.
I)xy is Even*Odd is Even. II) x-y is Even+/-Odd is Odd. III) 3x is Odd*Even =Even, 2y is Even*Odd=Even, Even + Even = Even. Therefore only II is Odd.
← Didn't Know|Knew It →
If x represents an even integer, which of the following expressions represents an odd integer?
If x represents an even integer, which of the following expressions represents an odd integer?
Tap to reveal answer
Pick any even integer (2, 4, 6, etc.) to represent x. The only value that is odd is 3_x_ + 1. Any number multiplied by an even integer will be even. When an even number is added and subtracted to that product, the result will be even as well. 3_x_ + 1 is the only choice that adds an odd number to the product.
Pick any even integer (2, 4, 6, etc.) to represent x. The only value that is odd is 3_x_ + 1. Any number multiplied by an even integer will be even. When an even number is added and subtracted to that product, the result will be even as well. 3_x_ + 1 is the only choice that adds an odd number to the product.
← Didn't Know|Knew It →
The sum of three consecutive odd integers is 93. What is the largest of the integers?
The sum of three consecutive odd integers is 93. What is the largest of the integers?
Tap to reveal answer
Consecutive odd integers differ by 2. If the smallest integer is x, then
x + (x + 2) + (x + 4) = 93
3x + 6 = 93
3x = 87
x = 29
The three numbers are 29, 31, and 33, the largest of which is 33.
Consecutive odd integers differ by 2. If the smallest integer is x, then
x + (x + 2) + (x + 4) = 93
3x + 6 = 93
3x = 87
x = 29
The three numbers are 29, 31, and 33, the largest of which is 33.
← Didn't Know|Knew It →
, where 
and 
are distinct positive integers. Which of the following could be values of 
and 
?
Tap to reveal answer
Since 
and 
must be positive, eliminate choices with negative numbers or zero. Since they must be distinct (different), eliminate choices where 
. This leaves 4 and 5 (which is the only choice that does not add to 20), and the correct answer, 5 and 15.
Since
← Didn't Know|Knew It →
You are given that 
are all positive integers. Also, you are given that:
is an odd number. 
can be even or odd. What is known about the odd/even status of the other four numbers?
You are given that
Tap to reveal answer
The odd/even status of 
is not known, so no information can be determined about that of 
.
is known to be an integer, so 
is an even integer. Added to odd number 
, an odd sum is yielded; this is 
.
is known to be odd, so 
is also odd. Added to odd number 
, an even sum is yielded; this is 
.
is known to be even, so 
is even. Added to odd number 
; an odd sum is yielded; this is 
.
The numbers known to be odd are 
and 
; the number known to be even is 
; nothing is known about 
.
The odd/even status of
The numbers known to be odd are
← Didn't Know|Knew It →
You are given that 
are all positive integers. Also, you are given that:
is an odd number. 
can be even or odd. What is known about the odd/even status of the other four numbers?
You are given that
Tap to reveal answer
A power of an integer takes on the same odd/even status as that integer. Therefore, without knowing the odd/even status of 
, we do not know that of 
, and, subsequently, we cannot know that of 
. As a result, we cannot know the status of any of the other values of the other three variables in the subsequent statements. Therefore, none of the four choices are correct.
A power of an integer takes on the same odd/even status as that integer. Therefore, without knowing the odd/even status of
← Didn't Know|Knew It →
If x is an even number, y is an odd number, and z is an even number, which of the following will always give an even number?
I. xyz
II. 2x+3y
III. z2 – y
If x is an even number, y is an odd number, and z is an even number, which of the following will always give an even number?
I. xyz
II. 2x+3y
III. z2 – y
Tap to reveal answer
I. xyz = even * odd * even = even
II. 2x + 3y = even*even + odd*odd = even + odd = odd
III. z2 – y = even * even – odd = even – odd = odd
Therefore only I will give an even number.
I. xyz = even * odd * even = even
II. 2x + 3y = even*even + odd*odd = even + odd = odd
III. z2 – y = even * even – odd = even – odd = odd
Therefore only I will give an even number.
← Didn't Know|Knew It →
You are given that are all positive integers. Also, you are given that:
You are given that 
is odd, but you are not told whether 
is even or odd. What can you tell about whether the values of the other four variables are even or odd?
You are given that
You are given that
Tap to reveal answer
, the product of an even integer and another integer, is even. Therefore, 
is equal to the sum of an odd number 
and an even number 
, and it is odd.
, the product of odd integers, is odd, so 
, the sum of odd integers 
and 
, is even.
, the product of an odd integer and an even integer, is even, so 
, the sum of an odd integer 
and even integer 
, is odd.
, the product of odd integers, is odd, so 
, the sum of odd integers 
and 
, is even.
The correct response is that 
and 
are odd and that 
and 
are even.
The correct response is that
← Didn't Know|Knew It →
Solve: 
Solve:
Tap to reveal answer
Add the ones digits:
Since there is no tens digit to carry over, proceed to add the tens digits:
The answer is 
.
Add the ones digits:
Since there is no tens digit to carry over, proceed to add the tens digits:
The answer is
← Didn't Know|Knew It →
At a certain high school, everyone must take either Latin or Greek. There are 
more students taking Latin than there are students taking Greek. If there are 
students taking Greek, how many total students are there?
At a certain high school, everyone must take either Latin or Greek. There are
Tap to reveal answer
If there are 
students taking Greek, then there are 
or 
students taking Latin. However, the question asks how many total students there are in the school, so you must add these two values together to get:
or 
total students.
If there are
← Didn't Know|Knew It →