Euclid shared the method of calculating the greatest common divisor and the least common multiple of two numbers in his work ‘Elements’ in 300 BC. This method is called the ‘Euclidean Algorithm’, but there are also easier calculation methods derived from it.
Generally, we can calculate the LCM or LCM if we want by separating the numbers into their prime factors and then choosing the appropriate factors according to the situation. Although it seems like they will not work in the real world, they can be used in many different sectors such as textiles. Let’s look at what EBOB and EKOK are and how they are calculated together.
What is EBOB?
Greatest common divisor refers to the greatest common divisor of two or more numbers. There are two popular methods that can be used to calculate. One is the Euclidean algorithm that Euclidean expresses in Elements, and the second is a somewhat easier and more frequently used method.
Let’s denote the GPA of any two numbers as ‘EBOB()’. The Euclidean algorithm is an algorithm that works on any two numbers. With the numbers a and b in hand, we can talk about three situations.
- 1. Case: the numbers a and b are equal (a=b)
- 2. Case: the number a is greater than the number b (a>b)
- 3. Case: b is greater than a (b>a)
In case of equality, since two numbers are equal, their greatest common divisor is themselves. That is, EBOB(a,b)=a=b.
In any other case, we divide the larger number by the smaller number, continuing to divide by the smaller number until the remainder is zero. Now when a is greater than b, if the remainder of the division of a by b and c and c are nonzero, this time we divide b by c and look at the remainder. If it is not zero, we continue the process by dividing this number by c. When the remainder is 0, the small divisor we find is the EBOB of the two initial numbers.
How is the EBOB calculated?
- Method #1: Calculation with Euclidean algorithm
- Method #2: Calculation by factoring
Method #1: Calculation with Euclidean algorithm:
- Step 1#: Divide start
- Step 2#: Divide until 0
Let’s go through an example. Let our numbers be 28 and 16.
Step 1#: Start by dividing
When we divide 28 by 16, the remainder is 12. Since it is nonzero, we continue
Step 2#: Continuing to divide until we reach 0
When we divide 16 by 12, the remaining number is 4, we continue because it is still greater than zero. In the last stage, the numbers we have are 12 and 4. In the part of them, we have 0 left. In this case, the number 4 becomes the EBOB of 28 and 16. So EBOB (28, 16)=4
Method #2: Factoring:
- Step 1#: Prime factorization
- Step 2#: Common
Step 1#: Prime factorization
Let’s first prime the numbers 28 and 16 we have.
Step 2#: Selecting the common prime factors and multiplying the common numbers found
Now let’s choose the prime factors that are common, both numbers have 2 numbers We see that they are in common. By multiplying these numbers, we get their greatest common divisor. Of course the answer would still be 4.
So what is ECOC?
Least common multiple allows us to find the least common multiple of two or more given numbers. While we can use prime factors in the calculation, we can also calculate LCM using the greatest common divisor of two numbers.
In the first method, we start by listing the prime factors. Then we write the common factors once and multiply by including the non-common factors and we get the result. In the second method, we multiply the two numbers we have and put them in the absolute value. Then, by dividing the result by the EBOB of these two numbers, we arrive at LMIC. Now let’s look at them in detail.
How is LPC calculated?
- Method #1: Calculating LCM using prime factors
- Method #2: Calculating LCM by expressing prime factors with exponents
Method #1: Prime factors Calculating LCM using:
- Step 1#: Prime factorization of 28 and 16
- Step 2#: Take one of the same factors and arrive at the result
Step 1#: Prime factorization of 28 and 16
We prime 28 and 16 in the same way as in the EBOB explanation
Step 2#: Getting one of the same multipliers and reaching the result
Now we will take one of the same multipliers, the black marked twos will be taken once since they are the factors of both numbers. Other multipliers will also be added directly. In this case, the LCC is calculated as (28, 16)=7x2x2x2x2=112.
The point we need to pay attention to in this method is that we add only one of the common prime factors to the multiplication. Since the 2’s in the black circle are also in the factors of both numbers, we add one of them to the process. Thus, we can get the result by multiplying 4 2 and one 7.
Method #2: Calculating LCM by expressing prime factors with exponents:
- Step 1#: Determining prime factors
- Step 2#: Arranging the determined numbers into exponents
- Step 3#: Multiply by the ones with the highest strength
This method is an easier version of the previous method. This time, after factoring the numbers into prime factors, we arrange them so that they become exponential numbers. Continuing from the example above:
Step 1#: Determining the prime factors
We factor 28 and 16 into prime factors in the same way as in the examples above.
Step 2#: Arranging the determined numbers in exponential numbers
After factoring into primes, we turn the numbers we have determined into exponential numbers.
Step 3#: Multiply by the ones with the highest strength
Here we will take the ones with the highest strength and multiply. So since 7 to the power is 1, we’re going to add 7 to the 1st power, and 2 to the highest power is 4, so we’re going to add 2 to the 4th product. When we do the operation, the answer is 112:
EBOB and LCM calculation tools:
We have stated above how to calculate the EBOB and LCM. Although there are ways we can do it ourselves, there are also applications that do these calculations for us. These applications are generally free to use and you can easily access most of them over the internet. In addition to websites, you can easily access applications that will calculate EBOB and LCC from your mobile devices.
PCC and LCC calculation tools for Andorid:
PCC and LCC calculation tools for iOS:
The apps you can find in the iOS store are unfortunately not as diverse as Android and the app itself is paid. But we still wanted to give it as an example.
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