Beginning Activity Beginning Acivity 2: The GCD and the Division Algorithm
When we speak of the quotient and the remainder when we âdivide an integer \(a\) by the positive integer \(b\text{,}\)â we will always mean the quotient \(q\) and the remainder \(r\) guaranteed by the Division Algorithm. (See Section 3.5, The Division Algorithm.)
1.
Each row in the following table contains values for the integers \(a\) and \(b\text{.}\) In this table, the value of \(r\) is the remainder (from the Division Algorithm) when \(a\) is divided by \(b\text{.}\) Complete each row in this table by determining \(\gcd( {a, b} )\text{,}\) \(r\text{,}\) and \(\gcd( {b, r} )\text{.}\)
| \(a\) | \(b\) | \(\gcd(a, b)\) | Remainder \(r\) | \(\gcd(b,r)\) |
| 44 | 12 | |||
| 75 | 21 | |||
| 50 | 33 |
2.
Formulate a conjecture based on the results of the table in Exercise 1.