Greatest Common Factor - ISEE Upper Level Quantitative Reasoning

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, .

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 .

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.

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:

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:

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:

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:

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 .

Find the factors of 72 and 80; the greatest common factor is the greatest number in both lists.

Fully factor and . Circle the factors in common:

Multiple the common factors to find the greatest common factor:

We are looking for a number that shares no factors with 10 other than 1. 20 and 28 can be eliminated since 2 divides 10, 20, 28. 25 and 35 can be eliminated since 5 divides 10, 25, and 25.

21 is the correct choice; and . Since they share no prime factors, , and 10 and 21 are relatively prime.

For a number to be relatively prime with , . Equivalently, and cannot share a prime factor. , so any number that does not have either or as a factor is a correct choice.

We can eliminate and , since is a factor of each; we can eliminate and , since is a factor of each. But , so . is the correct choice.

To find the greatest common factor (GCF), you need to determine the factor that both numbers share that is of the greatest value. List the factors of each number and identify the largest number in value that is in both lists:

The GCF of 20 and 36 is 4 since it is the largest number in value that shows up in both lists.

To find the greatest common factor (GCF), you need to determine the factor that all three numbers share that is of the greatest value. List the factors of each number and identify the largest number in value that is in all three lists:

The GCF of 15, 30, and 40 is 5 since it is the largest number in value that shows up in all three lists.

Factor each number:

Identify the common factors (). The greatest common factor is the product of all of the common factors.

The greatest common factor of two numbers is the largest shared factor that the two numbers have in common.

The factors of 36 are

The factors of 108 are

The largest shared factor between 36 and 108 is 36. Therefore, 36 is the correct answer.

Find the greatest common factor (GCF) of 56 and 64.

To find the GCF, first find the prime factors of our two numbers:

Now, to find our GCF, we need to find the prime factors our two numbers have in common.

In this case, we have three 2's in common. In other words, we have an 8 in common. So, our GCF is 8.

Find the greatest common factor (GCF) of 324 and 56.

The GCF is the largest number which will can be evenly divided from either number.

To find the GCF, first find the prime factors of both numbers. A good trick here is to start by pulling a two out of each number until you cannot pull out a two anymore.

Beginning with 56

So the prime factorization of 56 looks like:

Next, let's do 324.

So for 324 we get:

Now, the GCF can be found by taking the prime factors which are common to both numbers. Now, in this case, we have two 2's in each number, and nothing else. This means that the GCF of 56 and 324 is

So our answer is 4

Fully factor and . Circle the factors in common:

Multiple the common factors to find the greatest common factor: