### All PSAT Math Resources

## Example Questions

### Example Question #1 : How To Find The Greatest Common Factor

.

, , and are integers; they may or may not be distinct.

Which of the following cannot be equal to ?

**Possible Answers:**

**Correct answer:**

We look for ways to write 45 as the product of three integers, then we find the sum of the integers in each situation. They are:

Sum:

Sum:

Sum:

Sum:

Of the five choices, only 33 is not a possible sum of the factors. This is the correct choice.

### Example Question #2 : Factors / Multiples

What is the greatest common factor of and ?

**Possible Answers:**

**Correct answer:**

The greatest common factor is the largest factor that both numbers share. Each number has many factors. The factors for 72 are as follows:

Starting from the largest factor, 72, we can see that it is also a factor of 144

.

Therefore, 72 is the greatest common factor.

### Example Question #72 : Integers

, , and are positive two-digit integers.

The greatest common divisor of and is 10.

The greatest common divisor of and is 9.

The greatest common divisor of and is 8.

If is an integer, which of the following could it be equal to?

**Possible Answers:**

**Correct answer:**

The greatest common divisor of and is 10. This means that the prime factorizations of and must both contain a 2 and a 5.

The greatest common divisor of and is 9. This means that the prime factorizations of and must both contain two 3's.

The greatest common divisor of and is 8. This means that the prime factorizations of and must both contain three 2's.

Thus:

We substitute these equalities into the given expression and simplify.

Since and are two-digit integers (equal to and respectively), we must have and . Any other factor values for or will produce three-digit integers (or greater).

is equal to , so could be either 1 or 2.

Therefore:

or

### Example Question #1 : Prime Numbers

The sum of the first seven prime numbers divided by two is

**Possible Answers:**

61

33

29

58

24

**Correct answer:**

29

The first seven primes are 2, 3, 5, 7, 11, 13, and 17. Don't forget about 2, the smallest prime number, and also the only even prime! Adding these seven numbers gives a sum of 58, and 58/2 = 29.

### Example Question #1 : How To Find Out If A Number Is Prime

If is a prime number, what could also be prime?

**Possible Answers:**

**Correct answer:**

Plug in a prime number such as and evaluate all the possible solutions.

### Example Question #3 : Prime Numbers

How many integers between 2 and 20, even only, can be the sum of two different prime numbers?

**Possible Answers:**

**Correct answer:**

There are 8 possible numbers; 4,6,8,10,12,14,16,18.

One is not a prime number, so only 8, 10, 12, 14, 16, and 18 can be the sum of two different prime numbers.

### Example Question #4 : Prime Numbers

Define a series of consecutive prime numbers to be a series of numbers, each prime, in which there are no other prime numbers between them. These are not necessarily consecutive numbers themselves. For example, the numbers 5,7 and 11 are consecutive prime numbers, although they are not consecutive numbers.

If is the first number in a series of prime numbers, which of the following could not be the value of the **last** number in the series?

**Possible Answers:**

**Correct answer:**

The primes, in order, are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ...

We create a few series:

-> series length 2: 2,3

-> series length 3: 3,5,7

-> series length 5: 5,7,11,13,17

-> series length 7: 7,11,13,17,19,23,29

etc.

We can then see that, of the answers, only 47 and 31 remain possibly correct answers. Now we need to decide which of those two are impossible.

We could do another series, but the series has 11 terms requiring us to go further and further up. If we do this, we'll find that it terminates at 47, meaning that 31 must be the correct answer.

Another way, however, is to notice that 29 is the end of the series. Since 31 is the very next prime number, if we start on 11, the series that terminates in 31 would have to have length 7 as well. Every series after will thus end on a number larger than 31, meaning we will never finish on a 31.

### Example Question #1 : Prime Numbers

If is a prime number, how many factors does have?

**Possible Answers:**

**Correct answer:**

The value of , or , is the product of and , so it will be divisible by 1, *p*, *p ** *p*, and nothing else (we know that the *p*’s are not divisible because they are prime). Therefore *p*^{2} has exactly three factors.

(Alternatively, we can plug in any prime number for *p* and see how many factors *p*^{2} has. For example, if *p* is 3, then the factors of *p*^{2}, or 9, are 1, 3, and 9.)

### Example Question #1 : How To Find Out If A Number Is Prime

The sum of four consecutive integers is 210. Which one of these four integers is prime?

**Possible Answers:**

53

57

47

51

49

**Correct answer:**

53

Let *x* represent the smallest of the four numbers.

Then we can set up the following equation:

Therefore the four numbers are 51, 52, 53, 54. The only prime in this list is 53.

### Example Question #1 : How To Find Out If A Number Is Prime

Which of the following is equal to the sum of the five smallest prime numbers?

**Possible Answers:**

28

27

18

25

33

**Correct answer:**

28

It is important to know what a prime number is in order to answer this question. A prime number is defined as any positive integer that is divisible only by the number 1 and itself. For example, 17 is a prime number because its only factors are 1 and 17.

The first five prime numbers are 2, 3, 5, 7, and 11. Remember, 1 is not a prime number, and 2 is the only even prime.

Therefore, the sum of the five smallest prime numbers is 28.

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