WHAT IS THE GREATEST COMMON FACTOR OF 4284 AND 2520? - MATHS - 2025

The greatest common factor of 4284 and 2520 is 252. There are several methods to calculate this number. These methods do not depend on the chosen numbers, therefore they can be applied in a general way.

The concepts of greatest common divisor and least common multiple are closely related, as will be seen later.

With just the name you can tell what the greatest common divisor (or the least common multiple) of two numbers represents, but the problem lies in how this number is calculated.

It should be clarified that when speaking of the greatest common divisor of two (or more) numbers, only whole numbers are being mentioned. The same happens when the least common multiple is mentioned.

What is the greatest common divisor of two numbers?

The greatest common divisor of two numbers a and b is the largest integer that divides both numbers at the same time. It is clear that the greatest common divisor is less than or equal to both numbers.

The notation used to refer to the greatest common divisor of the numbers a and b is gcd (a, b), or sometimes GCD (a, b).

How is the greatest common divisor calculated?

There are several methods that can be applied to calculate the greatest common divisor of two or more numbers. Only two of these will be mentioned in this article.

The first is the best known and most used, which is taught in basic mathematics. The second is not as widely used, but it has a relationship between the greatest common divisor and the least common multiple.

- Method 1

Given two integers a and b, the following steps are carried out to calculate the greatest common divisor:

- Decompose a and b into prime factors.

- Choose all the factors that are common (in both decompositions) with their lowest exponent.

- Multiply the factors chosen in the previous step.

The result of the multiplication will be the greatest common divisor of a and b.

In the case of this article, a = 4284 and b = 2520. By decomposing a and b into their prime factors, we obtain that a = (2 ^ 2) (3 ^ 2) (7) (17) and that b = (2 ^ 3) (3 ^ 2) (5) (7).

The common factors in both decompositions are 2, 3 and 7. The factor with the lowest exponent must be chosen, that is, 2 ^ 2, 3 ^ 2 and 7.

Multiplying 2 ^ 2 by 3 ^ 2 by 7 gives the result 252. That is, GCD (4284.2520) = 252.

- Method 2

Given two integers a and b, the greatest common divisor is equal to the product of both numbers divided by the least common multiple; that is, GCD (a, b) = a * b / LCM (a, b).

As can be seen in the previous formula, to apply this method it is necessary to know how to calculate the least common multiple.

How is the least common multiple calculated?

The difference between calculating the greatest common divisor and the least common multiple of two numbers is that in the second step the common and uncommon factors with their greatest exponent are chosen.

So, for the case where a = 4284 and b = 2520, the factors 2 ^ 3, 3 ^ 2, 5, 7 and 17 must be chosen.

By multiplying all these factors, we obtain that the least common multiple is 42840; that is, lcm (4284.2520) = 42840.

Therefore, applying method 2, we obtain that GCD (4284.2520) = 252.

Both methods are equivalent and it will be up to the reader which one to use.

References

1. Davies, C. (1860). New university arithmetic: embracing the science of numbers, and their applications according to the most improved methods of analysis and cancellation. AS Barnes & Burr.
2. Jariez, J. (1859). Complete course of physical mathematical sciences I mechanics applied to the industrial arts (2 ed.). railway printing press.
3. Jariez, J. (1863). Complete course of mathematical, physical and mechanical sciences applied to the industrial arts. E. Lacroix, Editor.
4. Miller, Heeren, & Hornsby. (2006). Mathematics: Reasoning And Applications 10 / e (Tenth Edition ed.). Pearson Education.
5. Smith, RC (1852). Practical and mental arithmetic on a new plan. Cady and Burgess.
6. Stallings, W. (2004). Network security fundamentals: applications and standards. Pearson Education.
7. Stoddard, JF (1852). The practical arithmetic: designed for the use of schools and academies: embracing every variety of practical questions appropriate to written arithmetic with origional, concise, and analytic methods of solution. Sheldon & Co.