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Beginning Activity Beginning Acivity 2: The GCD and the Division Algorithm
A4US

When we speak of the quotient and the remainder when we “divide an integer a by the positive integer b,” 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. In this table, the value of r is the remainder (from the Division Algorithm) when a is divided by b. Complete each row in this table by determining gcd(a,b), r, and gcd(b,r).

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.