# SAT Math : How to find the greatest common factor

## Example Questions

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### Example Question #1 : Factors / Multiples

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 #4 : 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 #71 : 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

### Example Question #711 : Arithmetic

What's the greatest common factor of 6 and 8?

Explanation:

Greatest common factor is a common factor shared by two or more numbers. Both numbers are even, so let's divide both numbers by two. We get . These are prime numbers (factors of one and itsef) in which we are done. Anytime we have two prime numbers or one prime and one composite number, we are finished. So the greatest common factor is .

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

What's the greatest common factor of 4 and 8?

Explanation:

Greatest common factor is a common factor shared by two or more numbers.  is a multiple of , so let's divide  for both numbers. We get . We are finished as these are the basic numbers. So the greatest common factor is .

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

What's the greatest common factor of 19 and 27?

Explanation:

Greatest common factor is a common factor shared by two or more numbers.  is a prime number.  is a composite number. Since we have a prime and composite number, the greatest common factor is

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

What's the greatest common factor of 24 and 74?

Explanation:

Greatest common factor is a common factor shared by two or more numbers. Both numbers are even, so let's divide two for both numbers. We get .  We have  one prime and one composite number, so we are finished. The greatest common factor is .

### Example Question #5 : Greatest Common Factor

What's the greatest common factor of 18 and 243?

Explanation:

Greatest common factor is a common factor shared by two or more numbers. If you know the divisibility rule of  (sum of digits are divisible by ), then the answer is just  as the quotient is . We have a prime and composite number. However, if you don't and only know the divisibility rule of , then we can divide both numbers by  to get . We do it once more to get . Since we divided twice by , we multiply these factors and this is our greatest common factor of . Our answer is .

### Example Question #6 : Greatest Common Factor

What's the greatest common factor of 33 and 121?

Explanation:

Greatest common factor is a common factor shared by two or more numbers. If you know divisibility rule of , then this is the answer. However, this isn't easy to spot, so we will do process of elimination. The numbers are odd and if we have even factors, we never generate odd numbers so  is wrong. Next, check divisibility rule of . The digits of  add to  which isn't divisible by  so  is wrong. Next, let's divide  into . We get a decimal value and that's wrong since if we consider  to be a multiple of , it should be a whole number and not a decimal. Finally, by dividing  and , it's also . This is our answer. To find out if  is divisible by , just add the outside digits and match the middle one. Since it does,  is divisible by .

### Example Question #7 : Greatest Common Factor

What's the greatest common factor of 55 and 80?