Explaining The Least Common Denominator Of 15 And 12

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Introduction

As a professional teacher, it is my pleasure to explain the concept of the least common denominator (LCD) of 15 and 12. The LCD is a crucial concept in mathematics, especially in fractions. It is a value that is the least common multiple of the denominators of two or more fractions. In this case, we will focus on finding the LCD of 15 and 12.

Finding the Common Factors of 15 and 12

To find the LCD of 15 and 12, we need to first find the factors of each number. Factors are numbers that can be multiplied to produce another number. For example, the factors of 15 are 1, 3, 5, and 15. The factors of 12 are 1, 2, 3, 4, 6, and 12. We can see that both 3 and 12 are factors of 12 and 15.

Identifying the Greatest Common Factor

To determine the LCD, we need to find the greatest common factor (GCF) of 15 and 12. The GCF is the highest number that divides both 15 and 12 without leaving a remainder. To find the GCF, we can use the method of prime factorization. Prime factorization is the process of breaking down a number into its prime factors.

Prime Factorization of 15 and 12

The prime factorization of 15 is 3 x 5. The prime factorization of 12 is 2 x 2 x 3. We can see that both 3 and 12 are common factors of 15 and 12. The GCF of 15 and 12 is 3.

Calculating the Least Common Denominator

Now that we have found the GCF of 15 and 12, we can calculate the LCD. The LCD is the product of the GCF and the remaining factors of each number. In this case, we have 3 as the GCF, and the remaining factors of 15 are 5 and the remaining factors of 12 are 2 x 2.

Calculating the Product of GCF and Remaining Factors

To calculate the LCD, we multiply the GCF by the remaining factors of each number. 3 x 5 = 15 3 x 2 x 2 = 12

Comparing the Products

We can see that the products of the GCF and the remaining factors of each number are 15 and 12. The LCD is the least common multiple of these products.

Calculating the Least Common Multiple

To calculate the least common multiple, we need to find the smallest number that is divisible by both 15 and 12. We can use the method of listing multiples to find the least common multiple. Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, ... Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...

Comparing the Multiples

We can see that the least common multiple of 15 and 12 is 60. 60 is the smallest number that is divisible by both 15 and 12. Therefore, the LCD of 15 and 12 is 60.

Conclusion

In conclusion, the least common denominator (LCD) is a crucial concept in mathematics, especially in fractions. To find the LCD of 15 and 12, we need to find the greatest common factor (GCF) and calculate the product of the GCF and the remaining factors of each number. We can see that the least common multiple of 15 and 12 is 60, which is the smallest number that is divisible by both 15 and 12. By understanding the concept of the LCD, we can simplify fractions and perform operations on them more easily.