Important Definitions on the PSAT/NMSQT Math Sections
The arithmetic you learned in elementary and middle school serves you well when you’re working on numbers and operations questions on the PSAT/NMSQT. However, when you solve a problem on the PSAT/NMSQT, you may run into something like one of these phrases:
Three prime numbers added to . . .
The greatest positive number is . . .
A negative integer multiplied by . . .
You can’t do the problem if you don’t know what type of numbers you’re dealing with. Fortunately, the test-makers usually confine themselves to a few key terms.
An integer may be either positive (greater than zero) or negative (less than zero). Zero is also an integer, but it’s neither positive nor negative; it’s in a class by itself. Integers are never decimals or fractions.
A whole number is a positive number that never includes fractions or decimals. Whole numbers are even (divisible by 2) or odd (not evenly divisible by 2). Zero is also a whole number.
A prime number has only two factors; it can’t be divided by anything other than itself and 1. (In case you’re wondering, 1 and 0 are not prime numbers.)
A factor of a number is any number that divides neatly into another, larger, number without leaving a remainder. For example, 3 is a factor of 21, because when you divide 21 by 3, you get 7 and no remainder.
One more essential vocabulary word is consecutive (following one after another, with no interruption, as in “8, 9, 10”).
When you read a numbers and operations question, get in the habit of underlining the kind of number you’re searching for. Keep the type of number in your mind as you work through the problem and select an answer.
Check out these sample questions.
The product of three consecutive odd numbers is 315. What is the smallest of these integers?
Three prime numbers are multiplied together. Which of the following statements, if any, must be true?
I. The product must be odd.
II. The product must be prime.
III. The product must have exactly 5 factors.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) none of the above
What is the sum of the integers in the set
Now check your answers:
Plugging in is a great way to solve this problem. Remember, you want to try Choice (C) first. If 7 is the smallest number, then 9 and 11 are the other two numbers. Multiply those three together and you get 693 — much too big. Try Choice (B): 5 x 7 x 9 = 315, and you’ve found your answer!
Underlining key terms in the question is a great way to focus your attention on important details. In Question 1, you might underline “consecutive,” “odd,” “smallest,” and “integers.”
E. none of the above
How well do you know your prime numbers? Remember that 2 is the only even prime number, so if you multiply 2 by two other primes, the result is even. Therefore, Option I isn’t necessarily true. If you’re multiplying three numbers together to get your product, then each of those numbers is a factor of the product, so the product can’t be prime. Therefore, Option II is out.
And Option III is a trick! Pick three prime numbers to see what happens: 2, 3, and 5 will work, and their product is 30. You know that 2, 3, and 5 are all factors, but so is the product of any two of them: 6, 10, and 15. Also, remember that 30 and 1 are factors. Your answer is Choice (E).
You know that integers are positive or negative whole numbers, or 0. The integers in the set are –5, 0, and 10. When you add them together, the sum is 5, Choice (D).