Math 104A, Practice Final, Fall 2002.
All numbers are assumed to be integers unless otherwise stated.
1. The Fibonacci numbers fkare defined by f0= 0, f1= 1 and fk+1 =fk−1+fk.
Show by induction on dthat for d≥0 and k≥0,
fk+d+1 =fk+1fd+1 +fkfd.
2. Write down the general solution (x, y) of the equation 7x+ 8y= 50, and determine
all solutions with xand yboth positive.
3. If sand tare positive and relatively prime and rkis the remainder when ks is
divided by t, then
r0+· · · +rt−1=at2+bt +c.
What are the rational numbers a,band c? Explain.
4. List the numbers awith 0 ≤a < 20 such that there is at least one solution xto the
equation
ax ≡14 (mod 20).
5. Decide which of the following statements are true for every integer x, and all positive
integers cand m, and justify your answer.
(i) cx ≡0 (mod cm)⇒x≡0 (mod m).
(ii) x≡0 (mod m)⇒cx ≡0 (mod cm).
(iii) cx ≡0 (mod cm)⇒x≡0 (mod cm).
(iv) x≡0 (mod cm)⇒x≡0 (mod m).
6. Determine the prime factorization of 192 and find the smallest exponent esuch that
xe≡1 (mod 192) for every value of x.
* NOTE: This problem turns out to involve rather a lot of calculation. Such a
problem will not be on the real test!
7. Determine the number of solutions to the equation x16 ≡1 (mod 41), and the
number of solutions to the equation x16 ≡ −1 (mod 41).
8. Show that there are infinitely many primes of the form 6k+ 1.
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