Math 453 Homework 10 Due Wed., Nov. 28, 2007
Note the due date. This will count as a double assignment, but is actually about 1.5
times the length of recent assignments.
1. Strayer – Ch.5 – 13. (This is easier than #18 or #19.)
2. Strayer – Ch.5 – 30e.
3. Strayer – Ch.5 – 37.
4. (E) Find all solutions to the equation 4x+ 10y= 42 ...
a. ... In integers (x, y).
b. ... In positive integers (x, y).
5. (E) Find a primitive solutions to the Pythagorean equation x2+y2=z2for which (a)
x= 453, (b) x= 2008. (You do not have to multiply out your expressions, but make sure
they are primitive.)
6. (E) If integers (x, y, z) satisfy the equation x2+ 5y2=z2, then at least one of {x, y, z}
is divisible by 3. (Hint: Your proof (by contradiction) could begin: “Suppose neither x
nor yis divisible by 3”
7. (E) If integers (x, y, z) satisfy the equation x2+ 5y2=z2, then it is not true that at
least one of {x, y, z }is divisible by 7. (Hint: find a counterexample.)
8. (E) Find one solution to the Diophantine equation 400x2+ 53y2= 400z2in p ositive
integers (x, y, z). Observe that (x, y, z) = (1,0,1) is a solution to the equation but not
to the problem, because 0 is not a positive integer. This problem can be done in several
ways; you’re only asked for one.
9. (E) Each of these statements is either true, with a short proof, or false, with a simple
counterexample.
a. If (x, y , z) is a primitive solution to the equation x2+y2=z2, then x+y+z|xy.
b. ord31(2) = 30.
c. There do not exist positive integers xand zso that x2+ 1412=z2. (No restriction is
placed on gcd(x, z).)
10. (E) If w2+x2+y2=z2for integers (w, x, y, z ), prove that 6 |wxyz. (Example:
22+ 32+ 62= 72.)
11. (E) Suppose that gis a primitive root for an odd prime p≥7 and neither g3nor g5is
a primitive root mod p. Prove that p≡1mod 30.
12. Using any correct method, compute [2007]2,[2007]3,[2007]5, the base 2, base 3 and
base 5 representations of the current year.
13. Determine the largest integer rso that (6!)rdivides 2007!. (Hint: 6! = 24325.)
14. (Free problem) Happy Thanksgiving!