### All PSAT Math Resources

## Example Questions

### Example Question #1 : How To Subtract Even Numbers

If is an odd integer and is an even integer, which of the following must true of ?

**Possible Answers:**

The result will be odd.

The result will be even.

We cannot draw any conclusions from the given information.

**Correct answer:**

The result will be odd.

An even number subtracted from an odd number will always produce an odd result.

None of the other answer choices are correct.

### Example Question #1 : How To Add Even Numbers

If *x *represents an even integer, which of the following expressions represents an odd integer?

**Possible Answers:**

5*x* + 4

*x* + 2

2*x* – 2

3*x* + 1

3*x* – 2

**Correct answer:**

3*x* + 1

Pick any even integer (2, 4, 6, etc.) to represent *x*. The only value that is odd is 3*x* + 1. Any number multiplied by an even integer will be even. When an even number is added and subtracted to that product, the result will be even as well. 3*x* + 1 is the only choice that adds an odd number to the product.

### Example Question #1 : How To Divide Even Numbers

If *m* and *n* are both even integers, which of the following must be true?

l. *m*^{2}/*n*^{2} is even

ll. *m*^{2}/*n*^{2} is odd

lll. *m*^{2} + *n*^{2} is divisible by four

**Possible Answers:**

I only

II only

III only

I & III only

none

**Correct answer:**

III only

While I & II can be true, examples can be found that show they are not always true (for example, 2^{2}/2^{2} is odd and 4^{2}/2^{2} is even).

III is always true – a square even number is always divisible by four, and the distributive property tell us that adding two numbers with a common factor gives a sum that also has that factor.

### Example Question #1 : How To Divide Even Numbers

Let *S* be a set that consists entirely of even integers, and let *T* be the set that consists of each of the elements in *S* increased by two. Which of the following must be even?

I. the mean of *T*

II. the median of *T*

III. the range of *T*

**Possible Answers:**

I and II only

I only

II only

II and III only

III only

**Correct answer:**

III only

*S* consists of all even integers. If we were to increase each of these even numbers by 2, then we would get another set of even numbers, because adding 2 to an even number yields an even number. In other words, *T* also consists entirely of even numbers.

In order to find the mean of *T*, we would need to add up all of the elements in *T* and then divide by however many numbers are in *T*. If we were to add up all of the elements of *T*, we would get an even number, because adding even numbers always gives another even number. However, even though the sum of the elements in *T* must be even, if the number of elements in *T* was an even number, it's possible that dividing the sum by the number of elements of *T* would be an odd number.

For example, let's assume *T* consists of the numbers 2, 4, 6, and 8. If we were to add up all of the elements of *T*, we would get 20. We would then divide this by the number of elements in *T*, which in this case is 4. The mean of *T* would thus be 20/4 = 5, which is an odd number. Therefore, the mean of *T* doesn't have to be an even number.

Next, let's analyze the median of *T*. Again, let's pretend that *T* consists of an even number of integers. In this case, we would need to find the average of the middle two numbers, which means we would add the two numbers, which gives us an even number, and then we would divide by two, which is another even number. The average of two even numbers doesn't have to be an even number, because dividing an even number by an even number can produce an odd number.

For example, let's pretend *T* consists of the numbers 2, 4, 6, and 8. The median of *T* would thus be the average of 4 and 6. The average of 4 and 6 is (4+6)/2 = 5, which is an odd number. Therefore, the median of *T* doesn't have to be an even number.

Finally, let's examine the range of *T*. The range is the difference between the smallest and the largest numbers in *T*, which both must be even. If we subtract an even number from another even number, we will always get an even number. Thus, the range of *T* must be an even number.

Of choices I, II, and III, only III must be true.

The answer is III only.

### Example Question #1 : How To Multiply Even Numbers

If x is an even integer and y is an odd integer. Which of these expressions represents an odd integer?

I. xy

II. x-y

III. 3x+2y

**Possible Answers:**

I and II only

II and III only

II only

I and III only

I, II, and III only

**Correct answer:**

II only

I)xy is Even*Odd is Even. II) x-y is Even+/-Odd is Odd. III) 3x is Odd*Even =Even, 2y is Even*Odd=Even, Even + Even = Even. Therefore only II is Odd.

### Example Question #21 : Even / Odd Numbers

If x is an even number, y is an odd number, and z is an even number, which of the following will always give an even number?

I. xyz

II. 2x+3y

III. z^{2 }– y

**Possible Answers:**

II only

I and II only

I only

I, II and III

II and III only

**Correct answer:**

I only

I. xyz = even * odd * even = even

II. 2x + 3y = even*even + odd*odd = even + odd = odd

III. z^{2 }– y = even * even – odd = even – odd = odd

Therefore only I will give an even number.

### Example Question #1 : How To Multiply Even Numbers

If *x* and *y *are integers and at least one of them is even, which of the following MUST be true?

**Possible Answers:**

*x* + *y* is odd

*xy* is odd

*x* + *y *is even

*xy* is even

Nothing can be determined based on the given information

**Correct answer:**

*xy* is even

Since we are only told that "at least" one of the numbers is even, we could have one even and one odd integer OR we could have two even integers.

Even plus odd is odd, but even plus even is even, so *x* + *y* could be either even or odd.

Even times odd is even, and even times even is even, so *xy* must be even.

### Example Question #1551 : Sat Mathematics

Let *a* and *b* be positive integers such that *ab*^{2} is an even number. Which of the following must be true?

I. *a*^{2 }is even

II. *a*^{2}*b* is even

III. *ab* is even

**Possible Answers:**

II and III only

I and II only

I only

I, II, and III

II only

**Correct answer:**

II and III only

In order to solve this problem, it will help us to find all of the possible scenarios of *a*, *b*, *a*^{2}, and *b*^{2}. We need to make use of the following rules:

1. The product of two even numbers is an even number.

2. The product of two odd numbers is an odd number.

3. The product of an even and an odd number is an even number.

The information that we are given is that *ab*^{2} is an even number. Let's think of *ab*^{2} as the product of two integers: *a* and *b*^{2}.

In order for the product of *a* and *b*^{2} to be even, at least one of them must be even, according to the rules that we discussed above. Thus, the following scenarios are possible:

Scenario 1: *a* is even and *b*^{2} is even

Scenario 2: *a* is even and *b*^{2} is odd

Scenario 3: *a* is odd and *b*^{2} is even

Next, let's consider what possible values are possible for *b*. If *b*^{2} is even, then this means *b* must be even, because the product of two even numbers is even. If *b* were odd, then we would have the product of two odd numbers, which would mean that *b*^{2} would be odd. Thus, if *b*^{2} is even, then *b* must be even, and if *b*^{2} is odd, then *b*^{ }must be odd. Let's add this information to the possible scenarios:

Scenario 1: *a* is even, *b*^{2} is even, and *b* is even

Scenario 2: *a* is even, *b*^{2} is odd, and *b* is odd

Scenario 3: *a* is odd, *b*^{2} is even, and *b* is even

Lastly, let's see what is possible for *a*^{2}. If *a* is even, then *a*^{2 }must be even, and if *a* is odd, then *a*^{2} must also be odd. We can add this information to the three possible scenarios:

Scenario 1: *a* is even, *b*^{2} is even, and *b* is even, and *a*^{2} is even

Scenario 2: *a* is even, *b*^{2} is odd, and *b* is odd, and *a*^{2} is even

Scenario 3: *a* is odd, *b*^{2} is even, and *b* is even, and *a*^{2} is odd

Now, we can use this information to examine choices I, II, and III.

Choice I asks us to determine if *a*^{2} must be even. If we look at the third scenario, in which *a* is odd, we see that *a*^{2} would also have to be odd. Thus it is possible for *a*^{2} to be odd.

Next, we can analyze *a*^{2}*b*. In the first scenario, we see that *a*^{2} is even and *b* is even. This means that *a*^{2}*b* would be even. In the second scenario, we see that *a*^{2} is even, and *b* is odd, which would still mean that *a*^{2}*b* is even. And in the third scenario, *a*^{2} is odd and *b* is even, which also means that *a*^{2}*b* would be even. In short, *a*^{2}*b* is even in each of the possible scenarios, so it must always be even. Thus, choice II must be true.

We can now look at *ab*. In scenario 1, *a* is even and *b* is even, which means that *ab* would also be even. In scenario *2*, *a* is even and *b* is odd, which means that *ab* is even again. And in scenario 3, *a* is odd and *b* is even, which again means that *ab* is even. Therefore, *ab* must be even, and choice III must be true.

The answer is II and III only.

### Example Question #1 : How To Multiply Even Numbers

Let equal the product of two numbers. If , then the two numbers COULD be which of the following?

**Possible Answers:**

8 and 8

32 and 2

0 and 16

20 and 4

2 and 8

**Correct answer:**

2 and 8

The word "product" refers to the answer of a multiplication problem. Since 2 times 8 equals 16, it is a valid pair of numbers.

### Example Question #29 : Integers

If and is an odd integer, which of the following could be divisible by?

**Possible Answers:**

**Correct answer:**

If is an odd integer then we can plug 1 into and solve for yielding 13. 13 is prime, meaning it is only divisible by 1 and itself.

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