When writing up mathematics, make sure to embed the mathematics in
English-language explanations. Use the proofs and explanations in the
Notes on Number Theory and Cryptography as a guide to style. 

Preliminary work (not of hand-in quality):

  phi(21)=?

  prime factorization: 21=3*7

  phi(21) = (3 - 1)*(7 - 1)

        = 2*6
        
        = 12


Final write-up:

  We are asked to find the value of the Euler phi-function at 21.
Since 21 is the product of the two primes 3 and 7, and we know from
Exercise 3 of the "Notes on Number Theory and Cryptography" that for
distinct primes p and q we have phi(pq)=(p-1)(q-1), we find that

              phi(21) = (3 - 1)*(7 - 1) = 2*6 =12. 




Comments: 

1. Notice that the write-up includes a brief statement of the problem.

2. What is on the page can be read aloud and understood as correct
English prose. 

3. Relevant reasons are supplied to support the most important
steps. (What is relevant is often related to what the "point" of the
problem is--what has recently been learned that the problem is using
or illustrating.)

4. Special attention is paid to the key logical constructs
"if...then", "if and only if", "there exists", and "for every", as
well as the important words "and", "or", and "not". (How many of
these came up in this example?)

5. Why all this fuss?

  (a) We are not just trying to get an answer, but to understand the
process that produces the answer.

  (b) We have to convince others (for example, the grader) that our
reasoning is correct. 

  (c) It is nice to have a clear, complete, readable solution in hand
to consult later, for example when working on other problems or
reviewing for exams.

  (d) The discipline of writing something down clearly and completely
forces clear and complete understanding. First of all, it is a test of
understanding (if you can't get it down clearly in writing, you don't
really understand it). Second, the process of writing, criticizing,
and revising is actually one of the best tools for PRODUCING better
understanding.