lec0215

lec0215 - Basics of Number Theory We will say that an...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 non-zero 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 non-zero 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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Here 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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 7

lec0215 - Basics of Number Theory We will say that an...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online