Understanding GCD and LCM: A Step-by-Step Approach to Solving Problems

When dealing with number theory problems, particularly those involving the Greatest Common Divisor (GCD) and Least Common Multiple (LCM), it is crucial to understand the relationship between these two concepts. This article aims to guide you through solving a specific problem: finding the other number given the GCD, LCM, and one number. We'll explore the mathematical relationships that will help you find the solution and provide a thorough explanation.

The Problem at Hand

The problem statement is as follows:

The GCD of two numbers is 30 and their LCM is 900. One of the numbers is 60. Find the other possible number.

Using the Relationship Between GCD, LCM, and Numbers

The relationship between the GCD and LCM of two numbers (a) and (b) is given by the equation:

[text{GCD}(a,b) times text{LCM}(a,b) a times b]

Given:

(text{GCD}(a, b) 30) (text{LCM}(a, b) 900) (a 60)

We substitute these values into the equation:

[30 times 900 60 times b]

Calculating the Solution

First, we calculate the left-hand side of the equation:

[27000 60 times b]

To find (b), we divide both sides by 60:

[b frac{27000}{60} 450]

Therefore, the other number is 450.

Verification Using Prime Factorization

To verify the solution, we can use the prime factorization method:

The prime factorization of 60 is (2^2 times 3^1 times 5^1). The prime factorization of 450 is (2^1 times 3^2 times 5^2).

The GCD is the product of the lowest powers of the prime factors:

[text{GCD} 2^1 times 3^1 times 5^1 30]

The LCM is the product of the highest powers of the prime factors:

[text{LCM} 2^2 times 3^2 times 5^2 900]

Both checks confirm that the GCD is 30 and the LCM is 900, verifying that the other number is indeed 450.

Brute Force Solution Using J Programming Language

In the J programming language, we can find the other possible numbers using the following code:

m~./
90 180 450 900

The other possible numbers are 90, 180, 450, and 900.

Further Exploration and Conclusion

It is also worth noting that the third number must be a multiple of the GCD (30) and must divide the LCM (900). Therefore, the possible values of the third number can be determined by:

30k, where (k) is an integer and (30k) must divide 900. The divisors of 900 that are multiples of 30 are: 30, 60, 150, 300, 450, 900.

Since 60 and 150 are already given, the other possible numbers are 30, 90, 180, 300, 450, and 900. However, the problem specifically asks for the other number when one of the numbers is 60, which we have already determined to be 450.

By understanding the relationship between GCD, LCM, and prime factorization, you can solve similar problems efficiently. Prime factorization is a powerful tool in number theory and enhances your problem-solving skills. Practice with similar questions to deepen your understanding of these concepts.