What is the definition of greatest common factor?

Answer 1

GCF The greatest common factor (or GCF, also HCF, highest common factor) of a group of positive integers is the largest positive integer by which any of the numbers can be divided to yield another integer.

To find the GCF, list the factors of each of the numbers in a set, then locate the largest integer that is a factor for all of them. This is the greatest common factor.

Here's an example problem : What is the GCF of 24 and 40?

• The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
• The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40.

The factors common to both are 1, 2, 4, and 8. But the greatest is, of course, 8.

There can be no Greatest Common Factor (GCF) for a single number. The Greatest Common Factor (GCF) is the largest factor common to two or more given numbers. You can also see that it is a multiple of every common factor.

So rather than list ALL the factors of a number (which could be numerous), the GCF can be discovered as the product of all prime common factors. For example :

For the two numbers 168 and 210, the prime factors are found by division.

• 168 = 2 x 2 x 2 x 2 x 3 x 7
• 420 = 2 x 2 x 3 x 5 x 7

and the factors they have in common are 2, 2, 3, and 7. Multiplying yields the GCF 84. (We counted 2 twice because it appears twice in each of the numbers.)The GCF is the largest number that divides evenly into each of a given set of numbers.

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 is 1.

Th greatest common factor, the GCF, is the largest number that is a factor of all numbers given in a problem. The GCF can be no larger than the smallest number given in the problem. If no other number is a common factor, then the GCF wil be the number 1. To find the GCF, list of possible factors of the smallest number given (incude the number itself. Starting with the largest factor (which is the number itself), determine if it is also a factor of all given numbers. If so, that it is the GCF. If that factor doesn't work, try the same method on the next smaller factor. Continue until a common factor is found. That is the GCF.
A number that can be divided by a number or even if it could be divided by the factors.
In math, the greatest common divisor (gcd), also known as the greatest common factor (gcf), the greatest common denominator, or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.
The greatest common factor, or GCF, is the largest integer that divides evenly into two or more given numbers with no remainder.
The greatest common factor, or GCF, is the largest number that can be divided evenly into all the members of a given set of numbers.

Answer 2

It's the largest positive whole number that goes into all of them evenly.

Answer 3

Given a set of integers, their greatest common factor is the largest positive integer which goes into each of them evenly.

Answer 4

They are a characteristic of sets of two or more integers.

Answer 5

It is a common factor of two or more numbers which is greater than some other common factor.