# ISEE Upper Level Quantitative : How to find the greatest common factor

## Example Questions

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

3/5 + 4/7 – 1/3 =

Possible Answers:

3/37

7/9

88/105

72/89

Correct answer:

88/105

Explanation:

We need to find a common denominator to add and subtract these fractions. Let's do the addition first. The lowest common denominator of 5 and 7 is 5 * 7 = 35, so 3/5 + 4/7 = 21/35 + 20/35 = 41/35.

Now to the subtraction. The lowest common denominator of 35 and 3 is 35 * 3 = 105, so altogether, 3/5 + 4/7 – 1/3 = 41/35 – 1/3 = 123/105 – 35/105 = 88/105. This does not simplify and is therefore the correct answer.

### Example Question #2 : How To Find The Greatest Common Factor

25 is the greatest common factor of 175 and which of these numbers?

Possible Answers:

None of the answers are correct.

Correct answer:

Explanation:

Of the four numbers given, 25 is only a factor of 150, since all multiples of 25 end in the digits 25, 50, 75, or 00. To determine whether 150 is correct, we inspect the factors of 150 and 175:

Factors of 150:

Factors of 175:

Since 25 is the greatest number in both lists, .

### Example Question #3 : How To Find The Greatest Common Factor

is an odd prime.

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given.

(b) is equal.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

The greatest common factor of two numbers is the product of the prime factors they share; if they share no prime factors, it is .

(a) . Since  is an odd prime,  and  share no prime factors, and .

(b) , since  is prime. Since  is an even prime,  and  share no prime factors, and .

The quantities are equal since each is equal to .

### Example Question #1 : Numbers And Operations

Column A               Column B

The GCF of             The GCF of

45 and 120             38 and 114

Possible Answers:

The quantity in Column B is greater.

There is not enough info to determine the relationship.

The quantities are equal.

The quantity in Column A is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

There are a couple different ways to find the GCF of a set of numbers. Sometimes it's easiest to make a factor tree for each number. The factors that the pair of numbers have in common are then multiplied to get the GCF. So for 45, the prime factorization ends up being: . The prime factorization of 120 is: . Since they have a 5 and 3 in common, those are multiplied together to get 15 for the GCF. Repeat the same process for 38 and 114. The prime factorization of 38 is . The prime factorization of 114 is . Therefore, multiply 19 and 2 to get 38 for their GCF. Column B is greater.

### Example Question #1 : Factors / Multiples

What is the greatest common factor of  and ?

Possible Answers:

Correct answer:

Explanation:

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

: None

:  None

Taking these together, you get:

### Example Question #1 : Factors / Multiples

What is the greatest common factor of  and ?

Possible Answers:

Correct answer:

Explanation:

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

: None

: None

: None

Taking these together, you get:

### Example Question #2 : Factors / Multiples

What is the greatest common factor of  and ?

Possible Answers:

Correct answer:

Explanation:

To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:

Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:

: None

: None

: None

Taking these together, you get:

### Example Question #1 : Greatest Common Factor

Annette's family has  jars of applesauce. In a month, they go through  jars of apple sauce. How many jars of applesauce remain?

Possible Answers:

Correct answer:

Explanation:

If Annette's family has  jars of applesauce, and in a month, they go through  jars of apple sauce, that means  jars of applesauce will be left.

The first step to determining how much applesauce is left it to convert the fractions into mixed numbers. This gives us:

The next step is to find a common denominator, which would be 15. This gives us:

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

, , , , and are five distinct prime integers. Give the greatest common factor of and .

Possible Answers:

Correct answer:

Explanation:

If two integers are broken down into their prime factorizations, their greatest common factor is the product of their common prime factors.

Since , , , , and are distinct prime integers, the two expressions can be factored into their prime factorizations as follows - with their common prime factors underlined:

The greatest common factor is the product of those three factors, or .