RD Sharma solution class 8 chapter 5 Playing with numbers Exercise 5.3

Exercise 5.3

Page-5.30

Question 1:

Solve each of the following Cryptarithms:
$$\begin{gathered} \begin{array}{ccc}& 3& 7\\ +& A& B\end{array} \\ \overline{\begin{array}{ccc}& 9& A\end{array}} \end{gathered}$$

Answer 1:

$$\begin{gathered} \mathrm{Two} \mathrm{possible} \mathrm{values} \mathrm{of} \mathrm{A} \mathrm{are}: \\ \left(\mathrm{i}\right) \mathrm{If} 7 + \mathrm{B} \le 9 \\ \therefore 3 + \mathrm{A} = 9 \\ \Rightarrow \mathrm{A} = 6 \\ \mathrm{But} \mathrm{if} \mathrm{A} = 6, 7 + \mathrm{B} \mathrm{must} \mathrm{be} \mathrm{larger} \mathrm{than} 9. \mathrm{Hence}, \mathrm{it} \mathrm{is} \mathrm{impossible}. \\ \left(\mathrm{ii}\right) \mathrm{If} 7 + \mathrm{B} \ge 9 \\ \therefore 1 + 3 + \mathrm{A} = 9 \\ \Rightarrow \mathrm{A} = 5 \\ \mathrm{If} \mathrm{A} = 5 \mathrm{and} 7 + \mathrm{B} = 5, \mathrm{B} \mathrm{must} \mathrm{be} 8 \\ \therefore \mathrm{A} = 5, \mathrm{B} = 8 \end{gathered}$$

Question 2:

Solve each of the following Cryptarithm:
$$\begin{gathered} \begin{array}{cc} A& B\\ +3& 7\end{array} \\ \overline{ \begin{array}{cc} 9 & A\end{array}} \end{gathered}$$

Answer 2:

$$\begin{gathered} \mathrm{Two} \mathrm{possibilities} \mathrm{of} \mathrm{A} \mathrm{are}: \\ \left(\mathrm{i}\right) \mathrm{If} \mathrm{B} + 7 \le 9, \mathrm{A} = 6 \\ \mathrm{But} \mathrm{clearly}, \mathrm{if} \mathrm{A} = 6, \mathrm{B} + 7 \ge 9; \mathrm{it} \mathrm{is} \mathrm{impossible} \\ \left(\mathrm{ii}\right) \mathrm{If} \mathrm{B} + 7 \ge 9, \mathrm{A} = 5 \mathrm{and} \mathrm{B} + 7 = 5 \\ \mathrm{Clearly}, \mathrm{B} = 8 \\ \therefore \mathrm{A} = 5, \mathrm{B} = 8 \end{gathered}$$

Question 3:

Solve each of the following Cryptarithm:
$$\begin{gathered} \begin{array}{ccc}& A& 1\\ +& 1& B\end{array} \\ \overline{\begin{array}{ccc}& B& 0\end{array}} \end{gathered}$$

Answer 3:

$$\begin{gathered} \mathrm{If} 1 + \mathrm{B} = 0 \\ \mathrm{Surely}, \mathrm{B} = 9 \\ \mathrm{If} 1 + \mathrm{A} + 1 = 9 \\ \mathrm{Surely}, \mathrm{A} = 7 \end{gathered}$$

Question 4:

Solve each of the following Cryptarithm:
$$\begin{gathered} \begin{array}{cccc}& 2& A& B\\ +& A& B& 1\end{array} \\ \overline{\begin{array}{cccc}& B& 1& 8\end{array}} \end{gathered}$$

Answer 4:

$$\begin{gathered} \mathrm{B} + 1 = 8, \mathrm{B} = 7 \\ \mathrm{A} + \mathrm{B} = 1, \mathrm{A} + 7 = 1, \mathrm{A} = 4 \\ \mathrm{So}, \mathrm{A} = 4, \mathrm{B} = 7 \end{gathered}$$

Question 5:

Solve each of the following Cryptarithm:
$$\begin{gathered} \begin{array}{cccc}& 1& 2& A\\ +& 6& A& B\end{array} \\ \overline{\begin{array}{cccc}& A& 0& 9\end{array}} \end{gathered}$$

Answer 5:

$$\begin{gathered} \mathrm{A} + \mathrm{B} = 9 \mathrm{as} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{digits} \mathrm{can} \mathrm{never} \mathrm{be} 19 \\ 2 + \mathrm{A} = 0, \mathrm{A} \mathrm{must} \mathrm{be} 8 \\ \mathrm{A} + \mathrm{B} = 9, 8 + \mathrm{B} = 9, \mathrm{B} = 1 \\ \mathrm{So}, \mathrm{A} = 8, \mathrm{B} = 1 \end{gathered}$$

Question 6:

Solve each of the following Cryptarithm:
$$\begin{gathered} \begin{array}{cccc}& A& B& 7\\ +& 7& A& B\end{array} \\ \overline{\begin{array}{cccc}& 9& 8& A\end{array}} \end{gathered}$$

Answer 6:

$$\begin{gathered} \mathrm{If} \mathrm{A} + \mathrm{B} = 8, \mathrm{A} + \mathrm{B} \ge 9 \mathrm{is} \mathrm{possible} \mathrm{only} \mathrm{if} \mathrm{A} = \mathrm{B} = 9 \\ \mathrm{But} \mathrm{from} 7 + \mathrm{B} = \mathrm{A}, \mathrm{A} = \mathrm{B} = 9 \mathrm{is} \mathrm{impossible} \\ \mathrm{Surely}, \mathrm{A} + \mathrm{B} = 8, \mathrm{A} + \mathrm{B} \le 9 \\ \mathrm{So}, \mathrm{A} + 7 = 9, \mathrm{Surely} \mathrm{A} = 2 \\ 7 + \mathrm{B} = \mathrm{A}, 7 + \mathrm{B} = 2, \mathrm{B} = 5 \\ \mathrm{So}, \mathrm{A} = 2, \mathrm{B} = 5 \end{gathered}$$

Question 7:

Show that the Cryptarithm $4\times \overline{AB}=\overline{CAB}$ does not have any solution.

Answer 7:

$$\begin{gathered} 0 \mathrm{is} \mathrm{the} \mathrm{only} \mathrm{unit} \mathrm{digit} \mathrm{number}, \mathrm{which} \mathrm{gives} \mathrm{the} \mathrm{same} 0 \mathrm{at} \mathrm{the} \mathrm{unit} \mathrm{digit} \mathrm{when} \mathrm{multiplied} \mathrm{by} 4. \mathrm{So}, \mathrm{the} \mathrm{possible} \mathrm{value} \mathrm{of} \mathrm{B} \mathrm{is} 0. \\ \mathrm{Similarly}, \mathrm{for} \mathrm{A} \mathrm{also}, 0 \mathrm{is} \mathrm{the} \mathrm{only} \mathrm{possible} \mathrm{digit}. \\ \mathrm{But} \mathrm{then} \mathrm{A}, \mathrm{B} \mathrm{and} \mathrm{C} \mathrm{will} \mathrm{all} \mathrm{be} 0. \\ \mathrm{And} \mathrm{if} \mathrm{A}, \mathrm{B} \mathrm{and} \mathrm{C} \mathrm{become} 0, \mathrm{these} \mathrm{numbers} \mathrm{cannot} \mathrm{be} \mathrm{of} \mathrm{two}-\mathrm{digit} \mathrm{or} \mathrm{three}-\mathrm{digit}. \\ \mathrm{Therefore}, \mathrm{both} \mathrm{will} \mathrm{become} \mathrm{a} \mathrm{one}-\mathrm{digit} \mathrm{number}. \\ \mathrm{Thus}, \mathrm{there} \mathrm{is} \mathrm{no} \mathrm{solution} \mathrm{possible}. \end{gathered}$$