HR - proof - state things [ 'tractably' , 'comprehensively']

Submitted by metaperl on Sat, 04/01/2006 - 11:33am.
I am reading p.4 of The Haskell Road and for a long time struggled with one part of a proof on that page. It was his comment about c, a divisor of n. ... there are natural numbers a and b with c = a * b, and also 1 < a and a < c. but then a divides n..." and that is what confused me. He did not directly show how it was true that a divides n just because c was the least divisor of n.

State all implications

So let's take our statement and explore each isolated fact: First, c is the least divisor of n tell us all of this:
• c * q = n
• 1 * n = n
• c > 1

And c not prime tells us all of this:

• 1 * c = c
• a * b = c, where a != 1

Now what threw me is when the author of the text said: "Then there are natural numbers a and b with c = a * b, and also 1 < a and a < c. but then a divides n..." and that is what confused me. He did not directly show how it was true that a divides n. But if you look above, we have

` c * q = n`
but also
`a * b = c`
. By substituing
`a * b`
for c, we have
` a * b * q = n`
which means that a divide n.