Question

What are the quotient and remainder when

a) 19 is divided by 7?

b) -111 is divided by 11?

c) 789 is divided by 23?

d) 1001 is divided by 13?

e) 0 is divided by 19?

f) 3 is divided by 5?

g) -1 is divided by 3?

h) 4 is divided by 1?

Short answer

(a) Quotient 2 and reminder 5

(b) Quotient -11 and reminder 10

(c) Quotient 34 and reminder 7

(d) Quotient 77 and reminder 0

(e) Quotient 0 and reminder 0

(f) Quotient 0 and reminder 3

(g) Quotient -1 and reminder 2

(h) Quotient 4 and reminder 0

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 such that \(0 \le \;r < d\).
• \(q\)Is called the quotient and \(T\)is called the remainder.

(a): Finding quotient and remainder when 19 is divided by 7

\(a = 19\)

\(d = 7\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = 19\\ = 14 + 5\\ = 2 \cdot 7 + 5\\ = 2\;d + 5\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 2\)

The reminder is the constant: \(r = 5{\rm{\;(with\;}}\;0 \le 5 < 7{\rm{\;)\;}}\).

Hence the Quotient and Remainder is 2 and 5.

Step 3(b): Finding quotient and remainder when -111 is divided by 11

\(a = - 111\)

\(d = 11\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = - 111\\ = - 121 + 10\\ = ( - 11) \cdot 11 + 10\\ = ( - 11)\;d + 10\end{array}\).

The quotient is then the coefficient of\(d \Rightarrow q = - 11\)

The reminder is the constant: \(r = 10\;({\rm{\;with}}\;{\rm{\;}}0 \le 10 < 11)\)

Hence the Quotient and Remainder is -11 and 10.

Step 4(c): Finding quotient and remainder when 789 is divided by 23

\(a = 789\)

\(d = 23\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = 789\\ = 782 + 7\\ = 34 \cdot 23 + 7\\ = 34\;d + 7\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 34\)

The reminder is the constant: \(r = 7{\rm{\;(with\;}}\;0 \le 7 < 23{\rm{\;)}}{\rm{.\;}}\)

Hence the Quotient and Remainder is 34 and 7.

Step 5(d): Finding quotient and remainder when 1001 is divided by 13

\(a = 1001\)

\(d = 13\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = 1001\\ = 77 \cdot 13\\ = 77 \cdot 13 + 0\\ = 77\;d + 0\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 77\)

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

Hence the Quotient and Remainder is 77 and 0.

Step 6(e): Finding quotient and remainder when 0 is divided by 19

\(a = 0\)

\(d = 9\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = 0\\ = 0 \cdot 19\\ = 0 \cdot 19 + 0\\ = 0\;d + 0\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 0\)

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

Hence the Quotient and Remainder is 0 and 0.

Step 7(f): Finding quotient and remainder when 3 is divided by 5

\(a = 3\)

\(d = 5\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = 3\\ = 0 \cdot 5 + 3\\ = 0\;d + 3\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 0\)

The reminder is the constant: \(r = 3{\rm{\;(with\;}}\;0 \le 3 < 5{\rm{\;)}}{\rm{.\;}}\)

Hence the Quotient and Remainder is 0 and 3.

Step 8(g): Finding quotient and remainder when -1 is divided by 3

\(a = - 1\)

\(d = 3\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = - 1\\ = - 3 + 2\\ = ( - 1) \cdot 3 + 2\\ = ( - 1)\;d + 2\end{array}\)

The quotient is then the coefficient of \(d \Rightarrow q = - 1\)

The reminder is the constant: \(r = 2{\rm{\;(with\;}}\;0 \le 2 < 3{\rm{\;)}}{\rm{.\;}}\)

Hence the Quotient and Remainder is -1 and 2.

Step 9(h): Finding quotient and remainder when 4 is divided by 1

\(a = 4\)

\(d = 1\)

Let us determine the quotient and reminder:

\(\begin{array}{c}a = 4\\ = 4 + 0\\ = 4 \cdot 1 + 0\\ = 4\;d + 0\end{array}\)

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

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

Hence the Quotient and Remainder is 4 and 0.