###### Asked in Math and ArithmeticFactoring and Multiples

Math and Arithmetic

Factoring and Multiples

# What two numbers are the greatest common factor of 9 and a sum of 99?

## Answer

###### Wiki User

###### November 17, 2010 10:20PM

I had this on my hw and it is 90, and 9 hope i helped :)

## Related Questions

###### Asked in Geometry

### What are the kinds of factor?

###### Asked in Algebra, Factoring and Multiples, Prime Numbers

### What are two numbers whose greatest common factor is 7 and whose sum is 105 such that the larger is twice the smaller and the smaller has only two prime factors?

Since one number is twice the other, the smaller number must be
the greatest common factor. Since the greatest common factor is 7,
that would make the other number 14. But, 7 is a prime number and
has only one prime factor. However, the larger number, 14, has two
prime factors. Also, the sum of the two numbers is 21, not 105. So,
the information in the problem does not have a solution.
Let us ignore the greatest common factor information. Let the
smaller number be x. That means the larger number is 2x.
x + 2x = 105 => 3x = 105 => x = 35.
The two numbers are 35 and 70. The greatest common factor is 35.
The smaller number, 35, has only two prime factors.

###### Asked in Algebra, Factoring and Multiples

### What is the trick to finding the greatest common factor?

The following answer describes four methods of finding the
greatest common factor, with examples, and several "tricks" or
shortcuts that can make it easier.
Method: Guess and Refine
Sometimes, you can look at two numbers and make a good guess
that you can refine.
Example 1: Find the greatest common factor of 45 and 50.
Because both numbers end in either a 5 or 0, you know that they
are both divisible by 5. If you divide both numbers by 5 and the
results have no common factors (except 1), 5 is the greatest common
factor.
45 ÷ 5 = 9
50 ÷ 5 = 10
Since 9 and 10 are consecutive numbers, they have no common
factors. Therefore, the greatest common factor is 5.
Example 2: Find the greatest common factor of 150 and 750.
Both numbers end in 50, so they are both divisible by 50. If you
divide both numbers by 50 and the results have another common
factor, you continue identifying common factors until you have a
pair without common factors.
150 ÷ 50 = 3
750 ÷ 50 = 15
Since 15 is divisible by 3, and 3 is divisible by 3, you have
another common factor, which is 3. Then, you can divide the most
recent results by 3.
3 ÷ 3 = 1
15 ÷ 3 = 5
Since 1 and 5 do not have any common factors, take the two
factors that you did identify, 50 and 3, and multiply them
together: 50 x 3 = 150. This number, 150, is the greatest common
factor.
Method: Find All the Factors
If the numbers are small enough or you know that they have only
a few factors, you can list all the factors of each number and
compare to determine the largest factor they have in common. One of
the related questions links will take you to a page with the
complete list of factors for numbers 1 through 100.
Example: Find the greatest common factor of 15 and 18.
The factors of 15 are 1, 3, 5, and 15.
The factors of 18 are 1, 2, 3, 6, 9, and 18.
The common factors are 1 and 3, so the greatest common factor is
3.
Example: Find the greatest common factor of 26 and 91.
The factors of 26 are 1, 2, 13, and 26.
The factors of 91 are 1, 7, 13, and 91.
The common factors are 1 and 13, so the greatest common factor
is 13.
Method: Find the Prime Factors
In situations where you cannot get a good start simply by
looking at the numbers, follow the following steps:
1. Determine the prime factors of each number. See the related
question "How do you find prime factors" for a method on doing
this. Also, one of the related questions links will take you to a
page with the complete list of prime factors for numbers 1 through
100.
2. Determine the prime factors they have in common.
3. Multiply all the prime factors they have in common to
calculate the greatest common factor. Example: Find the greatest
common factor of 5,544 and 37,620.
The prime factors of 5,544 are 2, 2, 2, 3, 3, 7, and 11.
The prime factors of 37,620 are 2, 2, 3, 3, 5, 11, and 19.
The common prime factors are 2, 2, 3, 3, and 11.
Therefore, the greatest common factor is 2 x 2 x 3 x 3 x 11 =
396. Example: Find the greatest common factor of 7,888 and
10,002.
The prime factors of 7,888 are 2, 2, 2, 2, 17, and 29.
The prime factors of 10,002 are 2, 3, and 1667.
The common prime factors are a single 2.
Therefore, the greatest common factor is 2. Method: Euclidean
Algorithm
This method is more efficient than finding the prime factors
when the numbers are large, but teachers might prefer that you gain
experience determining the prime factors of numbers. For this
method, divide the larger number by the smaller number, then divide
the "divisor" from the previous division by the remainder from the
previous division, and continue until a number divides evenly. That
divisor is the greatest common factor. Example: Find the greatest
common factor of 33 and 77.
77 ÷ 33 = 2 remainder 11
33 ÷ 11 = 3 with no remainder
So, the final divisor, 11, is the greatest common factor.
Example: Find the greatest common factor of 27 and 168.
168 ÷ 27 = 6 remainder 6
27 ÷ 6 = 4 remainder 3
6 ÷ 3 = 2 with no remainder
So, the final divisor, 3, is the greatest common factor.
---- Shortcut 1: If one number is a multiple of the
other, the smaller number is the greatest common factor, because it
is the largest possible factor of itself.
Example: Find the greatest common factor of 72 and 288.
288 is divisible by 72, therefore 72 is the greatest common
factor.
Shortcut 2: The greatest common factor of two numbers
cannot be larger than the difference between the two numbers. So,
you only need to test the numbers that are equal to or less than
the difference between those two numbers. Also, the greatest common
factor must be a factor of the difference between the two numbers.
(This shortcut can help with finding the greatest common factor of
three or more numbers. Examples are shown in the related question
on finding the greatest common factor of three or more
numbers.)
Example: Find the greatest common factor of 56 and 64.
The difference between 56 and 64 is 64 - 56 = 8. The largest
possible common factor is the difference itself. So, check whether
8 divides evenly into both of them.
56 ÷ 8 = 7
64 ÷ 8 = 8
Therefore, 8 is the greatest common factor. Example: Find the
greatest common factor of 72 and 88.
The difference between 88 and 72 is 88 - 72 = 16. Check whether
16 divides evenly into both of them. It does not. But, the greatest
common factor must be a factor of 16. The factors of 16 are 1, 2,
4, 8, and 16. So, try the next largest factor, 8, and see if it
divides evenly into both of them.
72 ÷ 8 = 9
88 ÷ 8 = 11
Therefore, 8 is the greatest common factor.
Example: Find the greatest common factor of 1003 and 1180.
The difference between 1180 and 1003 is 177. Check whether 177
divides evenly into both of them. It does not. But, the greatest
common factor must be a factor of 177. By using the divisibility
rule for 3, you know that 3 is a factor of 177, but the
divisibility rule indicates that neither 1003 nor 1180 are
divisible by 3. 177 ÷ 3 = 59, so check 59 as a factor of both
numbers. Note that 3 and 59 are both prime numbers, so they are the
only prime factors of 177, so if there is a greatest common factor
of 1003 and 1180 other than 1, since we have ruled out 177 and 3,
it must be 59.
1003 ÷ 59 = 17
1180 ÷ 59 = 20
Therefore, 59 is the greatest common factor. Corollary 1 to
Shortcut 2: If the numbers are only one number apart, they are
relatively prime and have no common factor other than 1. Example:
Find the greatest common factor of 4 and 5.
The difference is 1, so the greatest common factor is 1. They
are relatively prime.
Corollary 2 to Shortcut 2: If the difference between the
two numbers is 2 and the numbers are not even numbers, they are
relatively prime and have no common factor other than 1. If the
difference is 2 and they are both even, the greatest common factor
is 2.
Example: Find the greatest common factor of 13 and 15.
The difference is 2 and the numbers are not even, so the
greatest common factor is 1. Example: Find the greatest common
factor of 14 and 16.
The difference is 2 and the numbers are even, so the greatest
common factor is 2.
Corollary 3 to Shortcut 2: If the difference between the
two numbers is a prime number, either that number is the greatest
common factor or 1 is the greatest common factor. Example: Find the
greatest common factor of 40 and 69.
The difference is 29, which is a prime number. Since 29 does not
divide evenly into both 40 and 69, the greatest common factor is 1,
which means they are relatively prime. Example: Find the greatest
common factor of 91 and 104.
The difference is 13, which is a prime number. Since 13 divides
evenly into both 91 and 104, the greatest common factor is 13.
91 ÷ 13 = 7
104 ÷ 13 = 8 Shortcut 3: If one of the numbers is prime,
either it is the greatest common factor or the greatest common
factor is 1. (Its only factors are 1 and itself, so those are the
only possible common factors it could have with another
number.)
Example: Find the greatest common factor of 83 and 90.
83 is a prime number and it is not a factor of 90, so the
greatest common factor is 1. Example: Find the greatest common
factor of 41 and 246.
41 is a prime number and it is a factor of 246, so the greatest
common factor is 41.
246 ÷ 41 is 6
---- Divisibility Rules:
To determine the prime factors, it is sometimes helpful to use
the divisibility rules.
2: The number ends in 0, 2, 4, 6, or 8.
Examples: 14, 58, 100, 3336
3: The sum of the number's digits is divisible by 3.
Examples: 78 (7+8=15 which is divisible by 3), 114 (1+1+4=6
which is divisible by 3)
5: The number ends in 0 or 5.
Examples: 70, 195, 4860
7: The last digit doubled subtracted from the rest of the
number is divisible by 7 or is equal to 0.
Examples: 343 (3x2=6; 34-6=28 which is divisible by 7), 875
(5x2=10; 87-10=77 which is divisible by 7)
11: Start with the left-most digit, subtract the next
one, add the next one, subtract the next one, etc., and the final
result is divisible by 11 or is equal to 0.
Examples: 165 (1-6+5=0), 308 (3-0+8=11 which is divisible by
11), 1078 (1-0+7-8=0)
Prime Numbers: Prime factors are prime numbers. The first
25 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

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