Question

Evaluate these quantities.

a) 13 mod 3

b)−97 mod 11

c) 155 mod 19

d)−221 mod 23

Short answer

1. 1
2. 2
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 13 mod 3

\(\begin{array}{l}a = 13\\d = 3\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 13\\ = 12 + 1\\ = 4 \cdot 3 + 1\\ = 4d + 1\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 4\)

The remainder is the constant:\(r = 1\;({\rm{with}}\;0 \le 1 < 3)\)

\(13\;\bmod \;3\)is then equal to the remainder

\(13\;\bmod \;3 = 1\)

Hence the solution is 1.

(b) Evaluating the quantity of -97 mod 11

\(\begin{array}{l}a = - 97\\d = 11\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = - 97\\ = - 99 + 2\\ = ( - 9) \cdot 11 + 2\\ = ( - 9)d + 2\end{array}\)

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

The remainder is the constant:\(r = 2\;({\rm{with}}\;0 \le 2 < 11)\)

\(97\;\bmod \;11\)is then equal to the remainder

\(97\;\bmod \;11 = 2\)

Hence the solution is 2.

(c) Evaluating the quantity of 155 mod 19

\(\begin{array}{l}a = 155\\d = 19\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 155\\ = 152 + 3\\ = 8 \cdot 19 + 3\\ = 8d + 3\end{array}\)

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

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

\(155\;\bmod \;19\)is then equal to the remainder

\(155\;\bmod \;19 = 3\)

Hence the solution is 3.

(d) Evaluating the quantity of -221 mod 23

\(\begin{array}{l}a = - 221\\d = 23\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = - 221\\ = - 230 + 9\\ = ( - 10) \cdot 23 + 9\\ = ( - 10)d + 9\end{array}\)

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

The remainder is the constant:\(r = 9\;({\rm{with}}\;0 \le 9 < 23)\)

\( - 221\;\bmod \;2\)is then equal to the remainder

\( - 221\;\bmod \;23 = 9\)

Hence the solution is 9.