Question

Show that if \(a\) and \(b\) are positive integers then the greatest common divisor of \(a\) and \(b\) divides their least common multiple. When are the least common multiple and the greatest common divisor equal to each other.

Short answer

In summary, the GCD of two positive integers a and b divides their LCM, as shown by the equation LCM(a,b) = (a * b) / GCD(a,b). Furthermore, the LCM and GCD are equal when the two numbers a and b are the same, as shown by the equation GCD(a,b)^2 = a * b.

Step-by-step solution

Express LCM in terms of GCD and vice versa

To show that the GCD of \(a\) and \(b\) divides their LCM, we can use the formula that relates these two values: \(a \cdot b = \text{GCD}(a,b) \cdot \text{LCM}(a,b)\) Now, we can express LCM in terms of GCD: \(\text{LCM}(a,b) = \frac{a \cdot b}{\text{GCD}(a,b)}\) This equation makes it clear that the GCD of \(a\) and \(b\) divides their LCM, since \(\text{LCM}(a,b)\) is an integer.

Determine when LCM and GCD are equal

Now, we want to find when the LCM of \(a\) and \(b\) is equal to their GCD: \(\text{LCM}(a,b) = \text{GCD}(a,b)\) Using the equation from Step 1, we can substitute the LCM with the GCD: \(\frac{a \cdot b}{\text{GCD}(a,b)} = \text{GCD}(a,b)\) Rearrange the equation to: \(\text{GCD}(a,b)^2 = a \cdot b\) Now, we can use the properties of GCD and LCM: 1. If \(a\) and \(b\) are coprime, their GCD is 1 and the LCM would be their product. Thus, if \(a\) and \(b\) are coprime, their LCM and GCD can never be equal since \(1 \neq a \cdot b\), unless \(a\) or \(b\) are equal to 1. 2. If \(a = b\), then we can replace \(b\) with \(a\) in our equation: \(\text{GCD}(a,a)^2 = a \cdot a \implies \text{GCD}(a,a) = a\) So the GCD and LCM are equal when \(a = b\). Thus, the least common multiple and the greatest common divisor are equal to each other when the two numbers are equal.

Key concepts

Greatest Common Divisor

Understanding the greatest common divisor (GCD), also known as the greatest common factor (GCF), is essential in many areas of mathematics. It is the largest positive integer that divides two or more integers without leaving a remainder. For example, for the integers 20 and 30, their GCD is 10. One method to find the GCD is by listing out all the factors of the numbers and choosing the largest common one, but this can be inefficient for larger numbers.

Another approach is using the Euclidean algorithm, which is based on the principle that the GCD of two numbers also divides their difference. This process is repeated until the remainder is zero, and the last non-zero remainder is the GCD. An interesting fact is that the GCD also reveals something about the multiples of numbers—specifically, if the GCD of two numbers is one, they are said to be coprime, which means they have no common factors other than 1.

Least Common Multiple

Another fundamental concept is the least common multiple (LCM). The LCM of two or more integers is the smallest positive integer that is evenly divisible by all the numbers. For instance, given the numbers 4 and 6, their LCM is 12. To find the LCM, one standard method is to list the multiples of the numbers and choose the smallest one that appears in both lists. This method, however, can become cumbersome with larger numbers or more than two integers.

Alternatively, using the relationship between LCM and GCD can simplify the process. Specifically, you can calculate the LCM of two numbers by dividing the product of the numbers by their GCD, as shown in the textbook solution. This mathematical relationship is particularly useful because it connects two seemingly separate concepts and allows for efficient calculations using basic arithmetic operations.

Integer Factorization

Integer factorization dovetails with the concepts of GCD and LCM, as it involves breaking down a number into a set of other numbers, or factors, which when multiplied together equal the original number. For example, the factorization of 10 is 2 × 5. The prime factorization of a number, which is the breakdown of a number into prime numbers that multiply to the original number, is often more useful. We use this prime factorization to determine the GCD and LCM by comparing the prime factors of the given integers.

For the GCD, we look for the highest power of common prime factors, and for the LCM, we find the least common powers of all the prime factors involved. Understanding integer factorization is a building block for higher-level concepts in number theory and plays a critical role in cryptography, where large prime numbers are a key component.

Coprime Integers

Coprime integers, also known as relatively prime or mutually prime numbers, are a pair of numbers that have no common factors other than 1. Their GCD is always 1. For example, 8 and 15 are coprime. When two numbers are coprime, it implies that no integer larger than 1 can evenly divide both numbers. This concept often arises in fractions—two fractions can be reduced to their simplest form if their numerators and denominators are coprime.

Additionally, it's interesting to note that when working with coprime integers, their LCM becomes simply the product of the two numbers. This is because the only shared factor they have to multiply by is 1. As implied in the textbook solution we referred to, knowing whether numbers are coprime can inform us about the possible equality or relationship between their GCD and LCM, as coprime numbers help to illustrate these concepts by acting as a clear, simplified example.