Lesson: Challenging Problem Solving - 13t03

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)

The only odd/even combination that results in an odd sum is our last option, 1 odd and 1 even number. Conceptually, this makes sense - even numbers are multiples of 2, and therefore consist of pairs of numbers with no 'leftovers'. The sum of two even numbers will also consist of pairs of numbers with no 'leftovers'. An example will illustrate: 8 (4 pairs) + 2 (1 pair) = 10 (5 pairs).

Two odd numbers also result in even numbers when summed. This is because an odd number consists of pairs of numbers (essentially an even number) plus another number. The two additional numbers combine to create another pair: 3 (1 pair, plus 1) + 5 (2 pairs, plus 1) = 8 (4 pairs).

Applied to our question stem, this rule indicates that x must be odd and y must be even, or vice versa.