Math 345/645 - Weekly homework 1

Math 345/645 - Weekly homework 1
The problems on this assignment should be written up impeccably and turned in on Tuesday,
January 26. This assignment does NOT need to be typed. However, all future written homework assignments will need to be typed in LATEX. For a rubric describing how this assignment
will be graded, see the syllabus. A problem with a ∗ is extra-credit for undergraduates and
required for graduate students.
1. Suppose that n is a positive integer, a ≡ b (mod n), and d is a positive divisor of n. Prove
that a ≡ b (mod d).
2. Suppose that a and b are two positive integers that are relatively prime (that is, there is no
number d > 1 so that d|a and d|b). Prove that if e is a positive integer that divides 3a + 2b
and 2a + b, then e = 1.
3. ∗ How many pairs (a, b) of positive integers are there for which
1 1
+ = ?
a b
Prove that your answer is correct.