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Basics of Number Theory
We will say that an integer a divides an integer b evenly
without a remainder, like this: a  b. Essentially, this implies
that there exists an integer c such that b = ac. We will only
define
division
by
nonzero
integers.
Hence,
it
is
not
permissible to write a  0.
Here are some rules that division of integers follow. (Note, a, b
and c are always nonzero integers.)
1) 1  a
2) a  0
3) if a  b, and b  c, then a  c.
4) if a  b and b  a, then a = +b or a = b
5) if x = y + z, and we have a  y and a  z, then a  x as well.
6) if a  b and a  c, then we have a  bx + cy for all ints x and y.
An example of how we can use these rules is as follows.
Are there any integer solutions to the equation
5x + 10y = 132?
The answer is no. We know that 5 must divide 5x and it must
also divide 10y, thus 5  (5x+10y). This must mean that for
integer solutions to exist 5  132. But this is not the case. Thus,
there are no integer solutions to this equation.
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View Full DocumentHere is another example of how to use these rules:
Prove that if 13 divides 3x+4y, that 13 also divides 7x+5y. We
can rewrite 7x+5y as 13x + 13y – 2(3x+4y). So we have:
7x + 5y = 13(x+y) – 2(3x + 4y)
Let A = x+y, B=3x+4y
We have 7x+5y = 13A – 2B
Since 13  13 and 13  B, it follows that 13  (13A – 2B). Since
7x+5y is equal to this value, it follows that 13  (7x + 5y).
Now, I will show that there are an infinite number of primes.
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 Spring '09

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