Example: Applying GCDs

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Example Applying GCDs

• The greatest common divisor (GCD) of two positive
  integers a and b is the largest integer that
  evenly divides both a and b.
• Mathematicians say that an answer like 3/6 is
  wrong, because it can be reduced to1/2 by
  dividing out by the GCD of the numerator and
  denominator - 3 in this case.
• An algorithm that computes the GCD of two numbers
  is useful in computer science for things as
  diverse as encryption for online-banking to
  controlling the motions of robots.

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Euclideans GCD A Loopy Algorithm

• Today's best known algorithm for solving this
  problem is the Euclidean Algorithm, based on
  the mathematics known to the ancient Greek world.
  
• Euclids basic process (algorithm) for computing
  the GCD of two numbers a and b is this
• Assuming a is the larger of the two,
  compute the remainder when a is
  divided by b, and
  replace a with b
  and b by the remainder.
• Keep doing this until you get a
  remainder of zero, at which point
  b is the GCD.
• For example, if a is 42 and b is 30, then 6 is
  the GCD.

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Coding Up Euclids Algorithm

• We can do this with a loop, and hopefully it's
  clear that the loop will look something like
  this
• while(not exactly sure what goes here! )
• r a b // remember "" is remainder
• a b
• b r
• Two questions still need to be answered
• When do we stop?
• Where is the answer when we stop?

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When to Stop Looping

• Our clue to stop is that r is zero (or, you could
  say that b being zero is our clue).
• Note that the answer resides in the variable a.
  So, our completed loop looks like this
• while (b ! 0)
• r a b // remember "" is remainder
• a b
• b r
• cout ltlt a ltlt " is the GCD!" ltlt endl

5
Example An Expensive Calculator

• Let's consider writing a program that reads from
  the user a simple arithmetic expression using
  and/or -, like
• 3 4 - 7 2 - 4
• Program prints out the result of the expression.
  Essentially your program will be a big while
  loop, and initially you might decide to write
  something like this

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Example An Expensive Calculator

• We can start filling in the parts of the code
  that we know how to do

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Example An Expensive Calculator

• Next well deal with reading in the /-'s, and
  using them to make up test conditions for our
  loop and if statements.
• Suppose we read the /- operation into a variable
  of type char which we'll call op.
  Three problems to
  face
• when to read in op
• how to do the loop
  test condition
• how to do the if test
  condition (lets tackle
  
  this first)

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Where to read in the value for op?

• The value for op will have to be read via cin gtgt
  op somewhere in the loop, but where?
• 3 choices

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Example Solution for Expensive Calculator

• include ltiostreamgt
• using namespace std
• int main()
• int total0, k
• char op ''
• coutltlt "Enter expression "

• while (op ! '') // Loop
• // read next value
• cin gtgt k
• // add or subtract val from total
• if (op '')
• total total k
• else
• total total - k
• cin gtgt op // read next op
• // end while loop
• // Write result
• cout ltlt total ltlt endl
• return 0
• // end main

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ICE Finding the Largest File

• Write a program that reads the below, and prints
  out the largest file size.
• File size (in bytes) appears in the column after
  "faculty".
• The berettagt at the end of the input indicates
  that there are no more file entries to read.