Assignment 2
Essential Arithmetic
MATH 335 Section 201
Due Friday, March 4th at 11 a.m.
1. Explain what it means for a ≡ b (mod n). Give some examples are part of your
explanation.
2. Use the multiplication table for Z5 to solve (i) 3x = 2 and (ii) 2x = 3:
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3. Consider the multiplication table for Z4 . Use it to explain why you can solve
3x = 2, and why you cannot solve 2x = 3.
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4. If today is Tuesday, March 1, 2011. What day of the week is March 1, 2012?
(Hint: Use modular arithmetic.)
5. Give an example to show that a2 ≡ b2 (mod n) need not imply that a ≡
b (mod n).
6. We use the notation n! to denote the product of all natural numbers less than
or equal to n. For example, 1! = 1, 2! = 1 · 2 = 2, 3! = 1 · 2 · 3 = 6, and so on.
These number grow at an incredible rate (e.g. 20! is a 19 digit number!)
(a) Compute 4!. What is 4! equal to mod 3?
(b) Compute 5!. What is 5! equal to mod 19?
(c) What is 1! + 2! + 3! + 4! + · · · + 99! + 100! equal to mod 12? (Hint: do not
try to compute this number first.)
7. Simplifying numbers modulo 7:
(a) Find the remainder when 260 is divided by 7.
(b) Find the remainder when 4165 is divided by 7.
(c) Show that the integer 111333 + 333111 is divisible by 7.
8. Suppose we are given the following UPC code, 10 digits in length: 5 2409 8082 1 .
We know that in order for the code to be valid, we must have the congruence
1 · d1 + d2 + 3 · d3 + d4 + 5 · d5 + d6 + 7 · d7 + d8 + 9 · d9 + d10 ≡ 0 (mod 10),
where d1 is the first (left-most) digit of the code, d2 is the second digit, and so
on. According to this definition, is the given code valid? If not, what should
the check digit be to give a valid UPC code?