Why would you leave an exam early?

As a student, and more so now as an instructor, it baffles me when I see students leave an examination early. Did they suddenly take ill? No. Did they get a perfect score? Negative. Is there some prize for leaving early? None. Then why did they leave?

Even when you think you've answered every question to the best of your ability, the remaining time can still be used to review your work, as many times as your time allows. Reviewing your work begins with the question, not with your solution. Reread the questions and ensure that they were interpreted correctly. Then review the solutions, redoing calculations where applicable. Mistakes can be made when using a calculator too! In some cases it is even possible to verify that the answer found is correct (e.g. by using another method or substituting the results to ensure that they satisfy all parts of the question.) Maybe in your haste your penmanship deteriorated. In this case you can use the time to rewrite parts of your solution more legibly.

Even when you draw a complete blank in an exam, it is in your best interest to stay. Take a few deep breaths and calm your nerves, then try the question again. You never know what you can remember unless you take the time to try to remember.

If you are a student who has left an exam early please share your reasons in the comments below.

Euclidean Algorithm

The Euclidean algorithm is a method to find the greatest common divisor of two positive integers, a and b, where a > b. We can denote the greatest common divisor of a and b as gcd(a,b)

The Euclidean algorithm is based on the following:

• If a and b are positive integers, there exist integers unique non-negative integers q and r so that a = bq + r , where r < b.

•  \(a = bq + r \rightarrow gcd(a,b) = gcd(b,r)\)

• gcd(z,0) = z, for any integer.

Algorithm steps:

1. Express a in the form a = bq+ r.
2. End if r = 0 and report b as the gcd. Otherwise b becomes our new a and r becomes our new b and repeat step 1.

Let's find the greatest common divisor of 732 and 345 using the Euclidean algorithm.

\begin{align*}
732 &= 345(2) + 42
\\345 &= 42(8) + 9
\\42 &= 9(4) + 6
\\9 &= 6(1) + 3
\\6 &= 3(2) + 0
\end{align*}

From above it follows that gcd(732,345) = gcd(345,42) = gcd(42,9) = gcd(9,6) = gcd(6,3) = gcd(3,0) = 3.


A common mistake is to report that the gcd(a,b) is the quotient, q, in the final line. In this case some might erroneously report gcd(732, 345) = 2. It helps to remember the second point above to avoid committing this mistake. The gcd is the last non-zero remainder.