Mathematics 312, Review Problems for Midterm 2
November 2015. Instructor: Z. Reichstein.
The midterm will cover sections 5.1, 5.5, 6.1, 6.2, 6.3, 7.1, 7.2, 7.3, 8.1, and 8.3 in the
text. Use odd-numbered problems from these sections (and the answer key in the back of
the book) for practice.
Here are a few additional problems.
Problem 1: Is the 10-digit integer n := 2222222222 divisible by (a) 3? (b) 5? (c) 9? (d)
11?
Problem 2: Find the last three decimal digits of 7999 .
Problem 3: Denote the integer 32760 = 23 · 32 · 5 · 7 · 13 by m. Prove that if gcd(a, m) = 1,
then a12 ≡ 1 (mod m).
Problem 4: Suppose n = p7 q 3 r5 , where p, q and r are distinct primes. How many positive
divisors does n have? What is the sum of these divisors? What is φ(n), where φ is the
Euler φ-function.
Problem 5: Prove that the number of positive divisors of n ≥ 1 is odd if and only if n is
a complete square.
Problem 6: Show that n = pi is not a perfect number for any prime p and any positive
integer i.
Problem 7: Recall that an affine cryptosystem, with encryption key KE = (26, a, b) and
decryption key KD = (26, c, d) works as follows. Each letter is assigned a numerical value,
in alphabetical order: A 7→ 0, B 7→ 1, . . ., Z 7→ 25. These numbers are then viewed modulo
26 and are encrypted by the formula
x 7→ y = ax + b (mod 26)
and decrypted by the formula
y 7→ cy + d (mod 26) .
It is known that the most frequently occurring letter in the English language is E (with
numerical value 4) and the second most frequently occurring letter is T (with numerical
value 19). Suppose these letters are encoded as F (numerical value 5) and G (numerical
value 6), respectively.
(a) Find the encryption key KE .
(b) Find the decryption key KD .