Lesson: Challenging Problem Solving - 13t07

Advanced Number Properties: Know The Rules

[Page 13 of 37]

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Picking Numbers works well on many Number Properties questions, but sometimes it's necessary to know Number Property rules in order to solve a problem. At other times, familiarity with the rules will help get you to the correct answer more efficiently. For the following Roman Numeral question, we will focus on the Number Property rules that are being tested. Review the question and then proceed to the task below.

If the sum of prime numbers x and y is an odd number, which of the following CANNOT be true?

I. y² is odd
II. x – y is odd
III. is an integer

	None
	I only
	III only
	I and II
	II and III

This question stem suggests that two Number Properties concepts will be central to solving this problem. What are they?

We have identified the two Number Properties concepts with which we're dealing. Now we must figure out how they apply to the question at hand. Our first rule involves odd/even number combinations. Which of the following odd/even combinations results in an odd number? Select all combinations that apply and then click Continue.

Which of the following odd/even combinations results in an odd number?

even + even

odd + odd

even + odd (or odd + even)

Our second rule involves prime numbers. What do we know about the properties of prime numbers? Use your knowledge of prime numbers to complete the rule below, and then click Continue.

is the only prime number; all other prime numbers are .

Now use what we've learned about x and y to evaluate each statement. Since each statement appears in the answer choices the same number of times, let's begin with the statement that seems easiest to handle. Statement II involves simple subtraction, so let's begin with this statement. Select the correct answer below, and then click Continue.

II. x – y is odd

		Statement II CAN be true
		Statement II CANNOT be true

This statement is very similar to the addition operation in the question stem. In fact, it too will result in an odd number. This makes sense - after all, subtraction and addition are related mathematical operations:

even – odd = odd even + (–odd) = odd
odd – even = odd odd + (–even) = odd

Note that we can now pick numbers to evaluate the statements. We only needed to know certain number property rules to get past the question stem. However, keep in mind that knowing the basic relationships among different types of numbers is a key skill that can get you through certain questions more efficiently than can picking numbers, and that may prove necessary on more difficult examples.

If we had picked 5 to represent the odd prime, we would have found that the sum is odd, and Statement II is always true:

If x = 5 and y = 2 x – y = 5 – 2 = 3
If y = 5 and y = 5 x – y = 2 – 5 = –3

We can eliminate all choices that contain Statement II — in this case Choices (D) and (E)