How to add odd numbers - PSAT Math
Card 1 of 63
, 
, 
, and 
are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of 
are odd.
II) Exactly three of 
are odd.
III) All four of 
are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of 
are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if 
and 
are odd, then the only odd number is 
). This makes 
the sum of one odd number and six even number and, subsequently, odd.
If exactly three of 
are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if 
, 
, and 
are odd, then the three odd numbers are 
, 
, and 
). This makes 
the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of 
are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes 
the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
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, , , and are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of are odd.
II) Exactly three of are odd.
III) All four of are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if and are odd, then the only odd number is ). This makes the sum of one odd number and six even number and, subsequently, odd.
If exactly three of are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if , , and are odd, then the three odd numbers are , , and ). This makes the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
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, , , and are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of are odd.
II) Exactly three of are odd.
III) All four of are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if and are odd, then the only odd number is ). This makes the sum of one odd number and six even number and, subsequently, odd.
If exactly three of are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if , , and are odd, then the three odd numbers are , , and ). This makes the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
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, , , and are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of are odd.
II) Exactly three of are odd.
III) All four of are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if and are odd, then the only odd number is ). This makes the sum of one odd number and six even number and, subsequently, odd.
If exactly three of are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if , , and are odd, then the three odd numbers are , , and ). This makes the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
← Didn't Know|Knew It →
, , , and are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of are odd.
II) Exactly three of are odd.
III) All four of are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if and are odd, then the only odd number is ). This makes the sum of one odd number and six even number and, subsequently, odd.
If exactly three of are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if , , and are odd, then the three odd numbers are , , and ). This makes the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
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, , , and are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of are odd.
II) Exactly three of are odd.
III) All four of are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if and are odd, then the only odd number is ). This makes the sum of one odd number and six even number and, subsequently, odd.
If exactly three of are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if , , and are odd, then the three odd numbers are , , and ). This makes the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
← Didn't Know|Knew It →
, , , and are positive integers.
is odd.
Which of the following is possible?
I) Exactly two of are odd.
II) Exactly three of are odd.
III) All four of are odd.
Which of the following is possible?
I) Exactly two of
II) Exactly three of
III) All four of
Tap to reveal answer
If exactly two of are odd, then exactly one of the seven expressions being added is odd - namely, the only one that does not have an even factor (for example, if and are odd, then the only odd number is ). This makes the sum of one odd number and six even number and, subsequently, odd.
If exactly three of are odd, then exactly three of the seven expressions being added are odd - namely, the three that do not include the even factor (for example, if , , and are odd, then the three odd numbers are , , and ). This makes the sum of three odd numbers and four even numbers and, subsequently, odd.
If all four of are odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makes the sum of seven odd numbers, and, subsequently, odd.
The correct choice is that all three scenarios are possible.
If exactly two of
If exactly three of
If all four of
The correct choice is that all three scenarios are possible.
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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.
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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.
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, 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
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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
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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
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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
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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
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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 →
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 →