### All PSAT Math Resources

## Example Questions

### Example Question #3 : Factors / Multiples

If is divisible by 2, 3 and 15, which of the following is also divisible by these numbers?

**Possible Answers:**

**Correct answer:**

Since *v* is divisible by 2, 3 and 15, *v* must be a multiple of 30. Any number that is divisible by both 2 and 15 must be divisible by their product, 30, since this is the least common multiple.

Out of all the answer choices, *v* + 30 is the only one that equals a multiple of 30.

### Example Question #1 : Factors / Multiples

Suppose that is an integer such that is ten greater than . What is the value of ?

**Possible Answers:**

**Correct answer:**

We are given information that m/4 is 10 greater than m/3. We set up an equation where m/4 = m/3 + 10.

We must then give the m variables a common denominator in order to solve for m. Since 3 * 4 = 12, we can use 12 as our denominator for both m variables.

m/4 = m/3 + 10 (Multiply m/4 by 3 in the numerator and denominator.)

3m/12 = m/3 + 10 (Multiply m/3 by 4 in the numerator and denominator.)

3m/12 = 4m/12 + 10 (Subtract 4m/12 on both sides.)

-m/12 = 10 (Multiply both sides by -12.)

**m = -120**

-120/4 = -30 and -120/3 = -40. -30 is 10 greater than -40.

### Example Question #1 : Factors / Multiples

; , , and are *distinct* integers.

Which of the following is a *possible* value of ?

**Possible Answers:**

**Correct answer:**

We need to find ways to factor 18 such that the three factors are different, and then find the sum of those factors in each case.

18 can be factored as the product of three integers in four ways:

I)

II)

III)

IV)

Disregard I and IV since each repeats a factor.

In (II), the sum of the factors is ; in (III) the sum is . Of the five choices, only 12 is possible.

### Example Question #71 : Integers

.

, , 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 #2 : Factors / Multiples

, , 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