Math 350 – Practice Exam 1

```C Roettger
Spring 2013
Math 350 – Practice Exam 1
Show all work, justify your conclusions. Calculators allowed, but no notes
or books. All the best!
Problem 1 Convert 2013 to binary and 1010001012 to decimal notation.
Problem 2 a) Find integers s, t such that
120s + 63t = gcd(120, 63)
b) Find the inverse of 32 modulo 43.
Problem 3 Find an integer x &lt; 120 such that
x≡4
(mod 8)
x≡2
(mod 3)
x≡1
(mod 5)
Problem 4 How many solutions does the congruence 12x ≡ 20 have modulo
40?
Problem 5 Why is 21000 − 1 not a prime? Find a short proof (NOT trial
division).
Problem 6 a) Explain why any number with base 16-representation n =
(a1 a2 . . . ak )16 satisfies
n ≡ a1 + a2 + &middot; &middot; &middot; + ak
(mod 15)
b) I wrote a number in base 16, n = A 46201B16 and unfortunately smudged
the second digit. But I know that n is a multiple of 15. What was the missing
digit?
Problem 7 Let fn be the n-th Fibonacci number (so f0 = f1 = 1, fn+2 =
fn+1 + fn ).
Prove by induction that gcd(fn+1 , fn ) = 1 for all n ≥ 1, and that the Euclidean
Algorithm finds this gcd in exactly n − 1 steps.
```