Beginning Activity Beginning Activity 1: The Greatest Common Divisor
1.
Explain what it means to say that a nonzero integer \(m\) divides an integer \(n\text{.}\) Recall that we use the notation \(m \mid n\) to indicate that the nonzero integer \(m\) divides the integer \(n\text{.}\)
2.
Let \(m\) and \(n\) be integers with \(m \ne 0\text{.}\) Explain what it means to say that \(m\) does not divide \(n\text{.}\)
Definition.
Let \(a\) and \(b\) be integers, not both 0. A common divisor of \(a\) and \(b\) is any nonzero integer that divides both \(a\) and \(b\text{.}\) The largest natural number that divides both \(a\) and \(b\) is called the greatest common divisor of \(a\) and \(b\text{.}\) The greatest common divisor of \(a\) and \(b\) is denoted by \(\gcd \left( {a, b} \right)\text{.}\)
3.
Use the roster method to list the elements of the set that contains all the natural numbers that are divisors of 48.
4.
Use the roster method to list the elements of the set that contains all the natural numbers that are divisors of 84.
5.
Determine the intersection of the two sets in Exercise 3 and Exercise 4. This set contains all the natural numbers that are common divisors of 48 and 84.
6.
What is the greatest common divisor of 48 and 84?
7.
Use the method suggested in Exercise 3 through Exercise 6 to determine each of the following: \(\gcd( {8, - 12})\text{,}\) \(\gcd( {0, 5} )\text{,}\) \(\gcd( {8, 27} )\text{,}\) and \(\gcd( {14, 28})\text{.}\)
8.
If \(a\) and \(b\) are integers, make a conjecture about how the common divisors of \(a\) and \(b\) are related to the greatest common divisor of \(a\) and \(b\text{.}\)