Description
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 control structure 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 that
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
531 3359
and
300 22355
, so clearly 3 divides
both numbers, while no number larger than 3 does. The problem with these methods 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 discovered 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:
a b q r
and
0 r b
. We call a the dividend, b the
divisor, q the quotient and r the remainder. If
r 0
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
531 3001 231.
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:
531 3001 231
Divide the preceding divisor by the preceding remainder:
300 2311 69
Divide the preceding divisor by the preceding remainder:
231 693 24
Divide the preceding divisor by the preceding remainder:
69 242 21
Divide the preceding divisor by the preceding remainder:
24 2113
Divide the preceding divisor by the preceding remainder:
21 37 0
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 also
that the process must eventually terminate, since on each iteration the remainder is less than the divisor
(0 ≤ 𝑟 < 𝑏), hence the sequence of remainders is strictly decreasing and must therefore reach zero.
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We illustrate on another example: find the GCD of 675 and 524. We begin by dividing the larger by the
smaller.
675 5241151
524151371
151 7129
71 97 8
9 811
8 18 0
We stop when we reach a remainder of 0, and conclude that the GCD of 675 and 524 is 1. (Check this
result by working out and comparing the prime factorizations of 675 and 524.) If you’d like to see proof of
the correctness of Euclid’s Algorithm, take Discrete Mathematics CMPE 16.
In this project you are to write a Java program that implements the Euclidean Algorithm described above.
It should be clear at this point that your program must use one of Java’s iterative control structures (i.e. a
while loop, do-while loop, or a for loop.) Recall that if a and b are int variables storing positive values,
then the expression
a % b
evaluates to the integer remainder of a upon division by b. Recall also that the
quotient is never used, so one need only declare int variables for the dividend, divisor, and remainder. On
each iteration you should compute the remainder from the dividend and divisor, update the values of
dividend and divisor, then go to the next iteration. The loop repetition condition should simply be that the
remainder is greater than zero.
In addition, your program will carry out a robust interaction with the user to check for correct input. In
particular, if the user enters anything other a positive integer, your program will continue to prompt for
more input. You must design appropriate loops to carry out these checks. Format your program output to
coincide with the sample runs below.
% java GCD
Enter a positive integer: 531
Enter another positive integer: 300
The GCD of 531 and 300 is 3
% java GCD
Enter a positive integer: 300
Enter another positive integer: 531
The GCD of 300 and 531 is 3
% java GCD
Enter a positive integer: -57
Please enter a positive integer: a;lsdkjf
Please enter a positive integer: 531
Enter another positive integer: qopweiru
Please enter a positive integer: 123.456
Please enter a positive integer: z.x,mvn
Please enter a positive integer: 300
The GCD of 531 and 300 is 3
Observe that the user may enter the numbers in any order, i.e. smaller—larger, or larger—smaller. Note
also that the program prompts separately for the two inputs, and that non-positive integers, double values,
and non-numeric strings are rejected until a positive integer is entered. It is recommended that you first
write a program that does no input checking of any kind, and just computes the GCD of two positive integers
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correctly. Only when this phase is complete, should you implement the input checks described above. See
examples on the website illustrating use of the break and continue commands for hints on how to
implement these checks.
What to turn in
Name your source file for this project GCD.java and submit it to the assignment name pa3.