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Euclidean Algorithm
It is not necessary to actually list the factors of all numbers to get the GCF for only two numbers. You can use the Euclidean algorithm.
(1) Divide the larger number by the smaller one.
(2) If there is no remainder, the GCF is the same as the smaller number.
(3) Repeat step 1 with the smaller number and the remainder.
Example:
GCF of 51 and 85.
85/51 = 1 R 34
51/34 = 1 R 17
34/17 = 2 R 0<== By step 2, we are done. Our answer is 17.
Lets try one that doesn't reduce -- GCF(17,39)
39/17 = 2 R 5
17/5 = 3 R 2
5/2 = 2 R 1
2/1 = 2 R 0, so our answer, the GCF, is 1.
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ViviIf the two numbers are the same then the GCF is the common value. Otherwise:
- find the difference between the larger number and the smaller number.
- replace the larger of the two numbers by this difference.
- if the two numbers are the same then that is the GCF, but if not go back to step 1.
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How do you use the greatest common factor subtraction method?
If you have two numbers A and B, and A > B, then GCF(A, B) = (A-B, B) Thus the problem of finding the GCF of A and B has been reduced to finding the GCF of B and a smaller number, A-B. This process can be continued until the two numbers are the same: and that number is the GCF.
How do you use prime factorization find the GCF of two numbers?
The first step of finding the GCF is to split the numbers into their prime factors. For instance, if I wanted to find the GCF of 30 and 105, I would split these up into: 30 = 2x3x5 105 = 3x5x7 The next step would be to identify any common prime factors. In this case both numbers have 3 and 5 as prime factors, so these would be the ones we use. To find the GCF, you simply multiply these two numbers together: 3x5 = 15 So 15 would be the GCF in that case.
What is the gcf of 180 and 1750?
The greatest common factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder. To find the GCF of 180 and 1750, you can use the prime factorization method. The prime factorization of 180 is 2^2 * 3^2 * 5, and the prime factorization of 1750 is 2 * 5^3 * 7. To find the GCF, you take the common prime factors with the lowest exponent, which in this case is 2 * 5 = 10. Therefore, the GCF of 180 and 1750 is 10.
What is the gcf of 75 100 and 175?
It's the same as gcf(gcf(75, 100), 175). In other words, you can first use Euclid's algorithm to find the gcf of 75 and 100; then you can calculate the gcf of the result with 175. To help you get started, by Euclid's algorithm, the gcf of 75 and 100 is the same as the gcf of 75 and 25 (where 25 is the remnainder of the division of 100 / 75).
How do use the distributive property for the gcf of 12 and 72?
It's not necessary. Since 12 is a factor of 72, it is automatically the GCF.
