# Ex.16.1 Q5 Playing with Numbers Solutions - NCERT Maths Class 8

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## 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.

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