# Ex.16.1 Q7 Playing with Numbers Solutions - NCERT Maths Class 8

## Question

Find the values of the letters in the following and give reasons for the steps involved.

\[\begin{align}{A \;\;{B}} \\ { \times \;\;\,{6}} \\ \hline B\,B\, {B} \\ \hline\end{align}\]

## Text Solution

**What is known?**

Multiplication operation of two numbers

**What is unknown?**

Value of alphabets i.e. \(A\) and \(B.\)

**Reasoning:**

Each letter in the puzzle must stand for just one digit. Each digit must be represented by just one letter.

**Steps:**

The multiplication of \(6\) and \(B\) gives a number whose one’s digit is \(B\) again.

It is possible only when

\(B = 0, \;2,\; 4,\; 6, \) or \(8\)

If \(B = 0,\) then the product will be \(0.\) Therefore, this value of \(B\) is not possible.

If \(B = 2\), then \(B \times 6 = 12\) and \(1\) will be a carry for the next step.

\(6{\text{A}} + 1 = {\text{BB}} = 22 \Rightarrow 6{\text{A}} = 21\) and hence, any integer value of \(A\) is not possible.

If \(B = 6,\) then \(B \times 6 = 36\) and \(3\) will be a carry for the next step.

\(6A + 3 = BB = 66 \Rightarrow 6A = 63\) and hence, any integer value of \(A\) is not possible.

If \(B = 8,\) then \(B \times 6 = 48\) and \(4\) will be a carry for the next step.

\(6{{A}} + 4 = {{BB}} = 88 \Rightarrow 6{{A}} = 84\) and hence, \(A = 14.\) However, \(A\) is a single digit number.

Therefore, this value of \(A\) is not possible.

If \(B = 4,\) then \(B \times 6 = 24\) and \(2\) will be a carry for the next step.

\(6A + 2 = BB = 44 \Rightarrow 6A = 42\) and hence, \(A = 7\)

The multiplication is as follows.

\[\begin{align}{7 \;\;{4}} \\ { \times \;\;\,{6}} \\ \hline 4\,4\, {4} \\ \hline\end{align}\]

Hence, the values of \(A\) and \(B\) are \(7\) and \(4\) respectively.

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