Finding the LCM (Least Common Multiple) of 75 and 80 involves several steps that need to be followed correctly to get the right answer. Let’s discuss each step in detail:
Step 1: Prime Factorization – The first step involves finding the prime factors of the given numbers. Prime factors are the smallest prime numbers that can divide a given number leaving no remainder. The prime factors of 75 and 80 are:
75 = 3 x 5^2
80 = 2^4 x 5
Step 2: Identifying Common and Uncommon Factors – Once the prime factorization is done, we need to compare the prime factors of both numbers to identify common and uncommon factors. Here, we can see that both 75 and 80 have the prime factor of 5, but only 75 has the prime factor of 3 and only 80 has the prime factor of 2.
Thus, the common factors are 5, and the uncommon factors are 3 and 2.
Step 3: Multiplying Common Factors – In this step, we take all the common factors identified in the previous step and multiply them. So, the product of the common factors of 5 is 5.
Step 4: Incorporating Uncommon Factors – Next, we need to include the uncommon factors in the LCM calculation. For this, we take the highest power of each uncommon factor and multiply them. Thus, we take 3 (highest power of 3) and 2^4 (highest power of 2). Then, we multiply the product from the previous step (common factors) with these uncommon factors to get the LCM.
So, LCM of 75 and 80 is:
LCM = 5 x 3 x 2^4 = 120
Therefore, the LCM of 75 and 80 is 120.
What is the LCM of 75 and 80 by prime factorization method?
To find the LCM of 75 and 80 by prime factorization method, we need to first factor both numbers into their prime factors.
Let’s start with 75:
75 = 3 x 5 x 5
Now let’s factor 80:
80 = 2 x 2 x 2 x 2 x 5
Next, we need to identify the common and non-common prime factors between the two numbers.
The common prime factor is 5, which appears in both factorizations.
The non-common prime factors in 75 are 3 and another 5, while the non-common prime factors in 80 are four 2’s.
To find the LCM, we take the highest exponent of each prime factors, whether they are common or non-common.
Therefore,
LCM = 2^4 x 3 x 5^2
LCM = 600
So, the LCM of 75 and 80 by prime factorization method is 600.
What is the greatest common factor of 75 and 80?
To find the greatest common factor (GCF) of 75 and 80, we need to identify the common factors of both numbers and then choose the greatest one. The factors of 75 are 1, 3, 5, 15, 25, and 75. The factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80. The common factors of 75 and 80 are 1 and 5.
To determine the greatest common factor, we choose the largest number that both 75 and 80 can be divided by evenly. In this case, that number is 5. Therefore, the greatest common factor of 75 and 80 is 5.
To verify this, we can divide 75 and 80 by 5. 75 ÷ 5 = 15 and 80 ÷ 5 = 16. Therefore, 5 is a factor of both 75 and 80, and there is no other number greater than 5 that can divide both of them. Hence, 5 is the greatest common factor of 75 and 80.
How do I find greatest common factor?
Finding the greatest common factor (GCF) of two or more numbers is essential when simplifying fractions, reducing square roots, or solving equations. The method for finding the GCF depends on whether the numbers are small or large or whether they are prime or composite. Here are some steps to help you find the GCF of two or more numbers:
Step 1: List the factors of each number. To list the factors of a number, you need to find all the numbers that divide into it without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Step 2: Identify the common factors. Look for the numbers that appear on both lists of factors. For example, the common factors of 12 and 18 are 1, 2, 3, and 6.
Step 3: Determine the greatest common factor. To find the GCF, you need to identify the largest number that the two numbers have in common. For example, the GCF of 12 and 18 is 6.
If you are dealing with more than two numbers, you can use the same method to find the GCF. After you have listed the factors of each number, identify the common factors, and then determine the largest one.
Another way to find the GCF is to use prime factorization. This method involves breaking down the numbers into their prime factors and then finding the factors that they have in common. For example, to find the GCF of 24 and 36, you would first list the prime factors of each:
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
Next, identify the common prime factors, which are 2 x 2 x 3 = 12. Therefore, the GCF of 24 and 36 is 12.
Finding the GCF involves identifying the common factors of two or more numbers and determining the largest one. You can use the listing method or the prime factorization method to accomplish this. Knowing how to find the GCF will help you simplify fractions, reduce square roots, and solve equations.
How to calculate LCM?
LCM stands for Least Common Multiple, which is the smallest number that is a multiple of two or more given numbers. Finding LCM is an important concept in mathematics, and it is frequently used in solving various problems related to numbers and arithmetic. There are different methods to calculate the LCM of two or more numbers, and some of the commonly used methods are:
Method 1: Prime Factorization
In this method, we need to find the prime factors of each number and then multiply the highest powers of each prime factor. Let’s take an example of finding LCM of 12 and 16:
Step 1: Prime Factorization of 12 = 2^2 x 3
Step 2: Prime Factorization of 16 = 2^4
Step 3: Now, we need to take the highest powers of each prime factor, i.e. 2^4 x 3 = 48.
Step 4: Therefore, the LCM of 12 and 16 is 48.
Method 2: Division Method
This method involves dividing the numbers by their common factors until we get the highest power of each factor. Let’s take the same example of finding LCM of 12 and 16:
Step 1: Divide 16 by 2, we get 8.
Step 2: Divide both 12 and 8 by 2, we get 6 and 4.
Step 3: Divide 6 by 2, we get 3.
Step 4: Since there are no more common factors, we multiply all the remainders and the divisors i.e. 2 x 2 x 2 x 3= 24.
Step 5: But this is not the LCM; we need to multiply by previously divided columns, i.e. 4 x 3 =12.
Step 6: Therefore, the LCM of 12 and 16 is 48.
Conclusion:
There are different methods available to calculate the LCM of two or more numbers, and each method has its own advantages and disadvantages. Understanding the concept of LCM is important for solving various problems related to numbers and arithmetic, and it is a fundamental concept in mathematics. By using the above-mentioned methods, anyone can easily calculate the LCM of two or more numbers.
