PSAT Math : How to add odd numbers

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 :Even / Odd Numbers

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

Possible Answers:

3d– 9

3d+ 4

3d– 3

3d+ 3

3d– 6

Correct answer:

3d– 6

Explanation:

If the largest of the three consecutive odd integers isd, 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 3d– 6.

Example Question #2 :Even / Odd Numbers

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

Possible Answers:

5 and 15

4 and 5

–10 and 30

0 and 20

10 and 10

Correct answer:

5 and 15

Explanation:

Since\dpi{100} pand\dpi{100} rmust 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.

Example Question #1 :Even / Odd Numbers

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

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #4 :How To Add Odd Numbers

你are given thatare 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?

Possible Answers:

ia odd;andare even;can be either.

None of the other responses are correct.

andare odd;andare even.

,, andare odd;can be either.

andare odd;is even;can be either.

Correct answer:

andare odd;is even;can be either.

Explanation:

The odd/even status ofis not known, so no information can be determined about that of.

is known to be an integer, sois an even integer. Added to odd number, an odd sum is yielded; this is.

is known to be odd, sois also odd. Added to odd number, an even sum is yielded; this is.

is known to be even, sois even. Added to odd number; an odd sum is yielded; this is.

The numbers known to be odd areand; the number known to be even is; nothing is known about.

Example Question #5 :How To Add Odd Numbers

你are given thatare 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?

Possible Answers:

,,, andare odd.

,,, andare even.

None of the other responses are correct.

andare even;andare odd.

andare odd;andare even.

Correct answer:

None of the other responses are correct.

Explanation:

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.

Example Question #6 :How To Add Odd Numbers

你are given thatare all positive integers. Also, you are given that:

你are given thatis odd, but you are not told whetheris even or odd. What can you tell about whether the values of the other four variables are even or odd?

Possible Answers:

andare even andandare odd.

andare odd;is even;can be either.

andare odd andandare even.

,,, andare odd.

andare even;is odd;can be either.

Correct answer:

andare odd andandare even.

Explanation:

, the product of an even integer and another integer, is even. Therefore,is equal to the sum of an odd numberand an even number, and it is odd.

, the product of odd integers, is odd, so, the sum of odd integersand, is even.

, the product of an odd integer and an even integer, is even, so, the sum of an odd integerand even integer, is odd.

, the product of odd integers, is odd, so, the sum of odd integersand, is even.

The correct response is thatandare odd and thatandare even.

Example Question #7 :How To Add Odd Numbers

,,, andare positive integers.

is odd.

Which of the following is possible?

I) Exactly two ofare odd.

II) Exactly three ofare odd.

III) All four ofare odd.

Possible Answers:

I and III only

I, II, and III

II and III only

None of I, II, or III

I and II only

Correct answer:

I, II, and III

Explanation:

If exactly two of奇怪,那么到底七表达式之一being added is odd - namely, the only one that does not have an even factor (for example, ifandare odd, then the only odd number is). This makesthe sum of one odd number and six even number and, subsequently, odd.

If exactly three ofare 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,, andare odd, then the three odd numbers are,, and). This makesthe sum of three odd numbers and four even numbers and, subsequently, odd.

If all four ofare odd, then all of the seven expressions being added, being the product of only odd numbers, are odd. This makesthe sum of seven odd numbers, and, subsequently, odd.

The correct choice is that all three scenarios are possible.

Example Question #8 :How To Add Odd Numbers

Solve:

Possible Answers:

Correct answer:

Explanation:

Add the ones digits:

Since there is no tens digit to carry over, proceed to add the tens digits:

The answer is.

Example Question #9 :How To Add Odd Numbers

在某一个high school, everyone must take either Latin or Greek. There aremore students taking Latin than there are students taking Greek. If there are普通外nts taking Greek, how many total students are there?

Possible Answers:

Correct answer:

Explanation:

If there are普通外nts taking Greek, then there areor普通外nts 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:

ortotal students.

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