### All PSAT Math Resources

## Example Questions

### Example Question #1 : How To Factor A Number

If x is the greatest prime factor of 42, and y is the greatest prime factor of 55, what is the value of xy?

**Possible Answers:**

21

10

77

15

105

**Correct answer:**

77

Find the prime factors of 42: 7, 3, 2

Find the prime factors of 55: 5, 11

Product of the greatest factors: 7 and 11 = 77

### Example Question #81 : Integers

If p is a prime number greater than 1, how many positive factors does p^{4} have?

**Possible Answers:**

Two

Five

Three

Six

Four

**Correct answer:**

Five

3 is a prime number that is easy to work with, so we can plug that in for p. p^{4} = 81. The positive factors of 81 are 1, 3, 9, 27, and 81. Thus, the answer is **five** factors.

Let's look at another prime number to plug in for p. If we plug in 2, another easy prime number to work woth, we get p^{4} = 16. The positive factors of 16 are 1, 2, 4, 8 and 16.

Notice that when p = prime number greater than 1, the positive factors for p^{4} are 1, p, p^{2}, p^{3} and p^{4}. Multiplying p^{n} by p only adds p^{n} as a factor when p is prime, so you will have n factors plus 1 which is a factor, so p^{4} has 5 factors.

### Example Question #1 : How To Factor A Number

What is the sum of all prime factors of 152?

**Possible Answers:**

25

6

21

9

**Correct answer:**

25

Since 152 is divisible by 2, start by dividing 152 by 2 which gives you factors of 2 and 76. 2 is a prime factor (cannot be divisible by anything other than itself and 1) but 76 can still be divided by 2. Continue dividing until there are only prime factors. You should get prime factors of 2, 2, 2, and 19. 2+2+2+19 = 25.

### Example Question #1 : How To Factor A Number

Let x, y, and z be three distinct positive integers. Which of the following values of y is possible,

if x^{3}y^{2}z = 1000

**Possible Answers:**

5

10

2,10

2,5

2

**Correct answer:**

2,5

If y = 2, then x^{3}z = 250. This means that x could be equal to 1, and z could be equal to 250. This would make x, y, and z distinct positive integers. Thus, it is possible for y to equal 2.

If y = 5, then x^{3}z = 40. This means x could be 1, and z could be 40. This would make x, y, and z distinct positive integers. Therefore, y can equal 5.

If y = 10, then x^{3}z = 10. This means that the only value that x could be is 1. If x were equal to 2, for example, then z would be equal to 10/8, which is not an integer. The same would be true if x were equal to anything other than 1. Thus, x must equal 1, and z must equal 10. However, because y and z must be distinct, y cannot equal 10.

Thus, y could equal only 2 and 5.

### Example Question #4 : Other Factors / Multiples

The product of three consecutive positive integers is 210. What is the sum of the integers?

**Possible Answers:**

**Correct answer:**18

Let x be the first integer. The next two integers would then be x+1 and x+2. We must find x such that:

x(x+1)(x+2) = 210.

To do this, we need to look at the factors of 210, and find a set of three factors that are also consecutive integers. First, let's find the prime factorization of 210, because then we can use that to determine all of the possible factors of 210.

210 = 10 x 21 = 2 x 5 x 21 = 2 x 5 x 3 x 7

Thus, the prime factorization of 210 is 2 x 3 x 5 x 7.

We can notice now that 5 and 7 are almost consecutive integers, and the only number missing between them is 6. If we multiply 2 and 3, we get 6. Thus 210 can be written as 5 x 6 x 7. This means that the consecutive integers are 5, 6, and 7, and their sum is 18.

### Example Question #5 : Other Factors / Multiples

If the variable x is an integer divisible by the numbers 2 and 3, which of the following is necessarily divisible by 2, 3 and 5?

**Possible Answers:**

5x + 30

x + 30

6x + 30

2x + 30

5x + 2

**Correct answer:**

5x + 30

For this question, use the fact that a sum of two multiples is a multiple. In other words:

if x is a multiple of 3 and y is a multiple of 3: (x + y) is a multiple of 3.

Thus in this question, x is a multiple of 2 and 3. We need to find a number that is a multiple of 2, 3, and 5.

Take 5x + 30:

[x is divisible by 2. 5 times x is still divisble by 3. 30 is divisible by 2.] -> divisible by 2.

[x is divisible by 3. 5 times x is still divisble by 3. 30 is divisible by 3.] -> divisible by 3.

[5x is divisible by 5. 30 is divisible by 5.] -> divisible by 5.

### Example Question #3 : How To Factor A Number

If *xy* = 100 and *x* and *y* are distinct positive integers, what is the smallest possible value of *x* + *y*?

**Possible Answers:**

29

101

25

20

52

**Correct answer:**

25

Consider the possible values for (*x*, *y*):

(1, 100)

(2, 50)

(4, 25)

(5, 20)

Note that (10, 10) is not possible since the two variables are to be distinct. The sums of the above pairs, respectively, are:

1 + 100 = 101

2 + 50 = 52

4 + 25 = 29

5 + 20 = 25, the smallest possible value.

### Example Question #1 : How To Factor A Number

If p is an integer, then which of the following could be equal to (6 + 2p)(3)

**Possible Answers:**

84

70

68

81

75

**Correct answer:**

84

Let n represent all of the numbers that are equal to (6 + 2p)(3). Then, let's solve for p in terms of n.

(6 + 2p)(3) = n

Divide both sides by 3.

(6 + 2p) = n/3

Subtract six from both sides.

2p = –6 + n/3

Divide both sides by 2.

p = –3 + n/6

Since we are told that p is an integer, the only way that p can be an integer is if n is a multiple of 6. Only a multiple of six, when divided by six, will yield an integer number. Any integer plus –3 will also be an integer, since the sum of two integers is always an integer.

In short, we must look for the answer choice that is a multiple of 6. Of the choices, only 84 is a multiple of 6.

The answer is 84.

### Example Question #4 : How To Factor A Number

What percentage of the positive factors of 48 are also factors of 168?

**Possible Answers:**

50

80

90

25

100

**Correct answer:**

80

First, we need to find all of the factors of 48. One way to do this is to come up with all of the pairs of positive integers that will multiply to yield 48. The pairs are as follows:

1 * 48

2 * 24

3 * 16

4 * 12

6 * 8

This means that the facors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Note that there are ten factors.

The question asks us for the percentage of the factors of 48 that are also factors of 168. This means we must see which of the ten factors 48 are also factors of 48. One way to do this is do divide 168 by each of the factors of 48 and see which give us whole numbers. If we divide 168 by a factor of 48 and get a whole number, then this number is also a factor of 168.

168/1 = 168

168/2 = 84

168/3 = 56

168/4 = 42

168/6 = 28

168/8 = 21

168/12 = 14

168/16 = 10.5

168/24 = 7

168/48 = 3.5

So, when we divided 168 by the factors of 48, we obtained a whole number, except when we divided by 16 and 48. This means that 16 and 48 are not factors of 168; however, 1, 2, 3, 4, 6, 8, 12, and 24 are all factors of 168.

Thus, out of the ten factors of 48, there are eight which are also factors of 168. We must express 8 out of 10 as a percent. 8/10 = 0.80 = 80%.

The answer is 80.

### Example Question #1 : How To Factor A Number

If there are to be parking spaces at least 6 feet wide along a wall 46 feet long, how many parking space can be drawn in?

**Possible Answers:**

7

5

4

6

8

**Correct answer:**

7

Only 7 hole spaces can fit. 8 * 6 = 48, which would be 2 feet longer than the wall.

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