# Ex.16.1 Q5 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.

\[\begin{align}{A \;{B}} \\ { \times \;\;{3}} \\ \hline C\, A\, {B} \\ \hline\end{align}\]

## Text Solution

**What is known?**

Multiplication operation of two numbers

**What is unknown?**

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

**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 \(3\) and \(B\) gives a number whose one’s digit is \(B\) again.

Hence, \(B\) must be \(0\) or \(5.\)

Let \(B\) be \(5.\)

Multiplication of first step \(= 3 × 5 = 15\)

\(1\) will be a carry for the next step.

We have, \(3 × A + 1 = CA\)

This is not possible for any value of \(A\).

Hence, \(B\) must be \(0\) only. If \(B = 0\), then there will be no carry for the next step.

We should obtain, \(3 \times A = CA\)

That is, the one’s digit of \(3 \times A\) should be \(A\). This is possible when \(A = 5\) or \(0.\)

However, \(A\) cannot be \(0\) as \(AB\) is a two-digit number.

Therefore, \(A\) must be \(5\) only. The multiplication is as follows.

\[\begin{align}{5 \;{0}} \\ { \times \;\;{3}} \\ \hline 1\,5\, {0} \\ \hline\end{align}\]

Hence, the values of \(A\),\(B\), and \(C\) are \(5, \,0, \) and \(1\) respectively.