Mathematical Preliminaries for COMP372 Algorithms

The modulus (or mod) function returns the remainder of an integer division. That is, given \(m, n \)are integers, \(n\) mod \( m \) is the positive integer \(r\) such that \(n = qm + r \) for q an integer and \(0 \leq r < m \). Alternatively, the modulus is \( n - m \lfloor \frac {n}{m} \rfloor. \) The result of n mod m must be between 0 and \(m-1\). For example, 5 mod 3 = 2, -3 mod 5 = 2, 5 mod 5 = 0, 30 mod 4 = 2, 567 mod 5 = 2.

1. Which of the following is untrue? \(m, n, p \) are integers.

\( (n+m) \) mod \(p \)= \( (n \) mod \( p \) + \( m \) mod \( p \) ) \(m \) mod 1 = 1 and \(m\) mod \(m \) = 0 \( (nm) \) mod \(p \) = \( ((n \) mod \( p\) )(\(m \) mod \(p\) )) mod \(p \) \( (n + m) \) mod \(p \) = \(n \) mod \(p \) + \(m \) mod \(p \)

2. Which answer is impossible for the following problem given any integer \(x\)? \( x \)mod 4

1 2 3 4

3. Solve the following equation for \(x\): \(x\) mod 13 = 1

1 14 27 There are infinite solutions to this equation.

4. You have \(x \) apples. You want to equally split your apples into 7 baskets. What expression will tell you how many apples you have left over when you do this?

7 mod \(x\) \(x\) mod 7 \(x\) - 7 ( \(x\) - ( \(x\) mod 7 ) ) / 7 Submit

Your grade is: __