Question

Evaluate these quantities.

a)−17 mod 2

b) 144 mod 7

c)−101 mod 13

d) 199 mod 19

Short answer

1. 1
2. 4
3. 3
4. 9

Step-by-step solution

Concept of Division Algorithm 

• Let\(a\)be an integer and\(d\)be a positive integer.
• Then there are unique Integers\(q\)and\(T\)with\(0 \le \;r < d\)such that\(a = dq + r\).
• \(q\)is called the quotient and\(T\)is called the remainder.
• \(\begin{array}{*{20}{l}}{q = a{\rm{ div }}d}\\{\;r = a{\rm{ mod }}d}\end{array}\)

(a) Evaluating the quantity of -17 mod 2

\(\begin{array}{l}a = - 17\\d = 2\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = - 17\\ = - 18 + 1\\ = ( - 9) \cdot 2 + 1\\ = ( - 9)d + 1\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = - 9\)

The remainder is the constant:\(r = 1\;({\rm{with}}\;0 \le 1 < 2)\)

\( - 17\;\bmod \;2\)is then equal to the remainder

\( - 17\;\bmod \;2 = 1\)

Hence the solution is 1.

(b) Evaluating the quantity of 144 mod 7

\(\begin{array}{l}a = 144\\d = 7\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 144\\ = 140 + 4\\ = 20 \cdot 7 + 4\\ = 20d + 4\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 20\)

The remainder is the constant:\(r = 4\;({\rm{with}}\;0 \le 4 < 7)\)

\(144\;\bmod \;7\)is then equal to the remainder

\(144\;\bmod \;7 = 4\)

Hence the solution is 4.

(c) Evaluating the quantity of 155 mod 19

\(\begin{array}{l}a = - 101\\d = 13\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = - 101\\ = - 104 + 3\\ = ( - 8) \cdot 13 + 3\\ = ( - 8)d + 3\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = - 8\)

The remainder is the constant:\(r = 3\;({\rm{with}}\;0 \le 3 < 13)\)

\( - 101\;\bmod \;13\)is then equal to the remainder

\( - 101\;\bmod \;13 = 3\)

Hence the solution is 3.

(d) Evaluating te quantity of 199 mod 19

\(\begin{array}{l}a = 199\\d = 19\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 199\\ = 190 + 9\\ = 10 \cdot 19 + 9\\ = 10d + 9\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 10\)

The remainder is the constant:\(r = 9\;({\rm{with}}\;0 \le 9 < 19)\)

\(199\;\bmod \;19\)is then equal to the remainder

\(199\;\bmod \;19 = 9\)

Hence the solution is 9.