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Name
INTRODUCTION TO NUMBER THEORY
Final Exam
May 8, 2000
This exam is worth 160 points. The point value of each problem is given in the margin.
(12) 1. Find all integer solutions of the linear equation 17x − 55 y = 1 with 200 < x < 300.
(6) 2. Find the order of 2 (mod 23). (Hint:
23
(6) 3. What is the “ones” digit of the number 7^999?
(12) 4. Let a = 2^125372 , b = 2^93457. Find the following.
(i) (a, b) = The prime factorization will do!
(ii) Find e such that 2e‖[a, b].
(iii) Find f such that 5f^ ‖(b − a).
(12) 5. Prove by induction that for any positive integer n the sum of the first n positive odd numbers is n^2 , that is
∑n k=1(2k^ −^ 1) =^ n
(12) 6. Find the least positive integer x satisfying the congruences
x ≡ 1 (mod 137) and 7x ≡ 2 (mod 13). (Don’t use trial and error.)
(12) 7. a) Define what it means for a function f defined on N to be multiplicative.
b) Suppose that f is a multiplicative function such that f (p) = −1, f (p^2 ) = 2 for any odd prime p and f (2) = 3. Calculate the following or state that it cannot be determined. f (30) =
f (18) =
f (54) =
(12) 12. Given that the only solution of x^3 + x + 1 ≡ 0 (mod 11) is x ≡ 2 (mod 11), solve the congruence x^3 + x + 1 ≡ 0 (mod 121). (Don’t use trial and error.)
(12) 13. Prove one of the following theorems: 1) There are infinitely many primes. 2) If n is a positive integer such that n has no prime divisor p with p <
n then n is a prime.
Do any three of the next six problems. Additional problems will count as extra credit at a value of 4 points per problem. (You’re best three will be worth 8 points each.)
(8) 14. Define the M¨obius function μ(n) and prove that it satisfies the identity
d|n μ(d) = 0 if^ n >^ 1.
(8) 15. Suppose that a, b are positive integers relatively prime to 10 such that the decimal expansions of 1/a and 1/b have repeating cycles of (minimal) lengths m, n respectively. If (a, b) = 1 what is the length of the repeating cycle for 1/(ab)? Prove your answer.
(8) 16. A right triangle is to be constructed such that the base has length 100 and the other two sides have integral lengths that are relatively prime to one another. How is this possible? (Give all possibilities.)
