# PSAT Math : How to find the greatest common factor

## Example Questions

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

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

Explanation:

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 #3 : Factors / Multiples

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

Explanation:

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 : How To Find The Greatest Common Factor

, and  are distinct integers.

Which of the following is a possible value of  ?

Explanation:

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 #2 : 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  ?

Explanation:

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 #3 : How To Find The Greatest Common Factor

What is the greatest common factor of  and ?

Explanation:

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 #75 : 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?

Explanation:

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