Understanding the Relation Between GCD and LCM

The greatest common divisor (GCD) and the least common multiple (LCM) of two numbers have a fundamental relationship. Specifically, the product of the GCD and LCM of two numbers is equal to the product of the numbers themselves. This relationship can be expressed as:

LCM(a, b) × GCD(a, b) a × b

This property is crucial for solving various number theory problems, particularly those involving the calculation of unknown numbers when their GCD and LCM are known.

Solving for the Second Number

In a specific problem, suppose we are given the GCD, LCM, and one of the two numbers. For example, if the GCD of two numbers is 7 and their LCM is 140, and one of the numbers is 20, we aim to find the other number.

Step by Step Solution

Using the given values:

Let's denote the two numbers as ( a ) and ( b ). Given: GCD 7, LCM 140, ( a 20 )

Applying the relationship:

LCM(a, b) × GCD(a, b) a × b

Substituting the values:

140 × 7 20 × b

Calculating the left side:

980 20 × b

To find ( b ), divide both sides by 20:

[ b frac{980}{20} 49 ]

Thus, the other number is 49.

Incorrect Approach and Why It Doesn't Work

One might initially think that the GCD could be 17 and the LCM could be 140. However, this approach is incorrect because the GCD must be a divisor of the LCM. Let's analyze why:

Checking the Divisibility

The GCD of 20 and 119 is not 17 since 20 is not divisible by 17. The LCM of 20 and 119 is not 140 because 119 does not have the necessary factors to form 140.

Therefore, the assumption that 17 could be the GCD is invalid.

Further Explorations

Exploring the relationship between GCD, LCM, and the numbers further:

HCF ( Highest Common Factor) of 20 and the other number should divide both 20 and 140. The factors of 20 are 1, 2, 4, 5, 10, 20. The only factor that divides both 20 and 140 is 20. The LCM of 20 and 70 is 140, as 70 is the smallest number that is a multiple of both 20 and 70.

Checking the factors, we see:

For 20: 1, 2, 4, 5, 10, 20 For 70: 1, 2, 5, 7, 10, 14, 35, 70

The LCM(20, 70) 140, satisfying the condition.

Conclusion and Application

The relationship between GCD and LCM is a powerful tool in number theory. Understanding and utilizing this relationship can solve many seemingly complex problems effortlessly. Always remember to check the divisibility and factorization properties to ensure the accuracy of your solution.

For more detailed information and further explorations into number theory, consider delving into the following advanced topics:

The relationship between GCD, LCM, and the prime factorization of numbers. Using Euclidean algorithms to find GCD. Exploring the least common multiple of more than two numbers.