Numbers & Operations
Tackling Numbers Game Items
As you’ll see, our step method is almost identical to Math and Supermath Heads. Follow these steps every time you see a Numbers Game item:
Step 1: Determine what the item is really asking you to solve.
Step 2: Determine which formulas and concepts to use.
Step 3: Do the math by writing out every step.
Step 4: Plug in numbers for variables to check your math.
Numbers Game Items in Slow Motion
We’ll look at this first Numbers Game item together:
7. If n and r are positive integers and (n + r)n is even, which of the following MUST be false?
(A) If n is odd, then r is odd.
(B) If n is odd, then r is even.
(C) If n is even, then r is even.
(D) If n is even, then r is odd.
(E) If n is zero, r must be even.
Step 1: Determine what the item is really asking you to solve.
The stem here wants you to apply your knowledge of odd/even number operations. What it really wants you to do is to discover which n and r combination makes the product odd, not even.
Step 2: Determine which formulas and concepts to use.
To solve this item, you should apply the formulas you just learned in the Essential Concepts section.
Remember the multiplication table for odd/even numbers? Here it is:
eveneven = even
evenodd = even
oddodd = odd
How about the addition table:
even + even = even
odd + odd = even
even + odd = odd
You definitely need to use these tables, so it is worth writing them down before you move ahead.
Step 3: Do the math by writing out every step.
From the multiplication table, you know that oddodd = odd. So both n and (n + r) should be odd.
To make the sum of n + r odd, either n or r must be odd. From our multiplication chart, we know that if n + r is odd, n must also be odd if we want the product of the two to be odd. Therefore, r should be even. Now go back to the answer choices. Answer B says that n is odd and r is even. This combination turns the answer into an odd number. Pick this answer choice.
Step 4: Plug in numbers for variables to check your math.
Now check your result with real numbers. If you don’t know the formulas or don’t trust your memory, you can always use real numbers to try out every answer offered or check the operations you performed.
Try out answer choice A: substitute n and r for odd numbers:
(3 + 5) 3 = 24
The answer is even and therefore should be eliminated.
Choice B: (3 + 2)3 = 15
The answer is odd, which contradicts the original condition. Choice B gets to hang around.
Choice C: (2 + 2)2 = 8
The answer is even and should be eliminated.
Plug in numbers for choices D and E, and you find out that answer B is the only combination that results in an odd number.
Guided Practice
Try this one on your own.
3. If abc = d and the value of a = 0, which of the following must be true?
(A) b = 0
(B) d = 0
(C) cb = 1
(D) b = 1
(E) d = 1
Step 1: Determine what the item is really asking you to solve.
What do you need to solve for to determine which of the answer choices is true?
Step 2: Determine which formulas and concepts to use.
You’ve got a zero here: what concepts can you use to figure this item out?
Step 3: Do the math by writing out every step.
Make sure you are using the concepts and formulas appropriately. Be careful not to mix different concepts together.
Step 4: Plug in numbers for variables to check your math.
Substitute real numbers into the answer choices to ensure you did everything right.
Guided Practice: Explanation
Step 1: Determine what the item is really asking you to solve.
In this item, we are dealing with a multiplication expression in which one of the numbers equals zero.
Step 2: Determine which formulas and concepts to use.
To verify/eliminate answer choices, you have to apply the rules of multiplication by zero. Specifically, the product of zero and a number is zero:
0n = 0
Step 3: Do the math by writing out every step.
Because a is zero, no matter what b and c are, the product d is always zero:
(0)bc = d
(0)bc = 0
Answer choice B is correct.
Step 4: Plug in numbers for variables to check your math.
Just to make sure there is only one possible answer to this item, check the equation with real numbers. Remember, the item asks you to choose the answer that must be true.
Check answer A: if abc = d and a is zero, b can equal zero. But because a = 0, b can be any number, because the product of a and b is zero anyway. Therefore, this answer is not necessarily true.
We already know answer choice B works.
Check answer C: if abc = d and a is zero, cb can equal 1. But because a = 0, the product of c and b can be any number, because the product of a, b, c is zero anyway. Therefore, this answer is not necessarily true.
You can work through answer choices D and E and find out that B is the only answer that must be true.
Independent Practice
Flip the page once you solve this item on your own.
7. If a is a multiple of 5, b is a multiple of 7, and ab = c, what is one possible value of c?
(A) 25
(B) 77
(C) 75
(D) 175
(E) 200
Independent Practice: Explanation
Step 1: Determine what the item is really asking you to solve.
You need to determine a value for c, given that a is a multiple of 5 and b is a multiple of 7.
Step 2: Determine which formulas and concepts to use.
This item is about multiples and factors. If 5 and 7 are factors of a and b, they both have to be factors of c.
Step 3: Do the math by writing out every step.
The simplest way to approach this item is to find the LCM shared by 5 and 7. Multiply the two numbers:
57 = 35
Can you reduce 35 to a smaller number that is divisible by both 5 and 7? Nope, so the correct answer must be divisible by 35.
Step 4: Plug in numbers for variables to check your math.
Try dividing each answer choice by 35. If you do your math carefully, you find that 175, choice D, is the correct answer.
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