Lesson: Challenging Problem Solving - 13t09

Advanced Number Properties: Know The Rules

[Page 13 of 37]

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

At this point it looks like we will need to evaluate both of the remaining statements to get the correct answer. Let's try Statement I next. Select the correct answer below, and then click Continue.

I. y² is odd

		Statement I CAN be true
		Statement I CANNOT be true

Since y can be either 2 or an odd prime, we must consider the result when 2, an even number, is squared, and the result when an odd number is squared. To evaluate this statement, we can rely on odd/even multiplication rules: