How to add odd numbers - PSAT Math

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Question

The sum of three consecutive odd integers is 93. What is the largest of the integers?

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.

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Question

Which of the following could represent the sum of 3 consecutive odd integers, given that d is one of the three?

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.

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Question

$\dpi{100} p+r=20$ , where $\dpi{100} p$ and $\dpi{100} r$ are distinct positive integers. Which of the following could be values of $\dpi{100} p$ and $\dpi{100} r$ ?

Answer

Since $\dpi{100} p$ and $\dpi{100} r$ must be positive, eliminate choices with negative numbers or zero. Since they must be distinct (different), eliminate choices where $\dpi{100} p=r$ . This leaves 4 and 5 (which is the only choice that does not add to 20), and the correct answer, 5 and 15.

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Question

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?

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 .

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Question

You are given that are all positive integers. Also, you are given that:

$A+ B^{2}= C$

$A+ C^{3}= D$

$A+ D^{4}= E$

$A+E^{5}= F$

is an odd number. can be even or odd. What is known about the odd/even status of the other four numbers?

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 $B^{2}$ , 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.

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Question

, , , and are positive integers.

is odd.

Which of the following is possible?

I) Exactly two of $\left { A, B,C, D\right }$ are odd.

II) Exactly three of $\left { A, B,C, D\right }$ are odd.

III) All four of $\left { A, B,C, D\right }$ are odd.

Answer

If exactly two of $\left { A, B,C, D\right }$ 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 $\left { A, B,C, D\right }$ 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 $\left { A, B,C, D\right }$ 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.

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Question

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?

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.

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Question

Solve:

Answer

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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Question

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?

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.

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