CMPS 12A
Introduction to Programming
Winter 2008
Programming Assignment 3
Due Thursday February 14, 10:00 pm
In this assignment you will write a java program that prompts the user for two positive integers, then
prints out the greatest common divisor of the two numbers. Your program will check its input, and will
repeatedly prompt the user until appropriate values are entered. The main program control construct used
in this program will be the loop (while, do-while, or for).
The greatest common divisor (GCD) of two positive integers is, as the name implies, the largest integer
which evenly divides both numbers. For instance the GCD of 531 and 300 is 3. One way to see this is to
list all divisors of both numbers, determine the set of common divisors, then determine the largest element
in that set.
Divisors of 531: {1, 3, 9, 59, 177, 531}
Divisors of 300: {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300}
Common Divisors of 531 and 300: {1, 3}
It follows that the GCD of 531 and 300 is 3. Another way to find the GCD is to simply compare the
prime factorizations of the two numbers. In this case
5933531
and
55322300
, so clearly 3
divides both numbers, while no number larger than 3 does. The problem with these techniques is that for
large integers, it is a very time consuming and computationally intensive task to determine the prime
factorization or the full set of divisors. Around 300 BC, Euclid of Alexandria published a very efficient
algorithm for the determination of the GCD. This algorithm is known universally today as the Euclidean
Algorithm. It uses the operation of integer division, which takes two positive integers as input, and
produces two integers as output.
Recall that if a and b are positive integers, then the quotient q and remainder r of a upon division by b are
uniquely determined by the two conditions:
rqba
and
br 0
. We call a the dividend, b the
divisor, q the quotient, and r the remainder. If
0r
we say that b divides evenly into a, or simply that b
divides a. (Note that the word “divisor” is used differently here than in the preceding paragraph, in which
it meant a number which divides evenly into another. The remainder need not be zero in general.) For
instance, dividing 531 by 300 yields a quotient of 1 and remainder of 231, since
2311300531
.
We now illustrate the Euclidean Algorithm on the two numbers 531 and 300.
Divide the larger number by the smaller, getting a quotient and remainder:
2311300531
Divide the preceding divisor by the preceding remainder:
691231300
Divide the preceding divisor by the preceding remainder:
24369231
Divide the preceding divisor by the preceding remainder:
2122469
Divide the preceding divisor by the preceding remainder:
312124
Divide the preceding divisor by the preceding remainder:
07321
The process halts when we reach a remainder of 0. The GCD is then the last non-zero remainder, which
is in this case is 3. Observe that on each iteration, we discard the dividend and quotient, and save only the
divisor and remainder for the next step, on which they become dividend and divisor respectively. Note
1
