# Ex.16.2 Q3 Playing with Numbers Solutions - NCERT Maths Class 8

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## Question

If $$24x$$ is a multiple of $$3,$$ where $$x$$ is a digit, what is the value of $$x$$?

(Since $$24x$$ is a multiple of $$3,$$ its sum of digits $$6+x$$ is a multiple of $$3$$ ,so $$6+x$$ is one of these numbers: $$0, 3, 6, 9, 12, 15, 18...$$ But since $$x$$ is a digit, it can only be that $$6+x=\,6$$ or $$9$$ or $$12$$ or $$15.$$ Therefore, $$x=0$$ or $$3$$ or $$6$$ or $$9.$$ Thus, $$x$$ can have any of four different values)

Video Solution
Playing With Numbers
Ex 16.2 | Question 3

## Text Solution

What is known?

A puzzled number

What is unknown?

Value of the alphabet i.e. $$x.$$

Reasoning:

If the sum of the digits of a number is divisible by $$3,$$ then the given number is a multiple of $$3.$$

Steps:

Since $$24x$$ is a multiple of $$3,$$ the sum of its digits is a multiple of $$3.$$

Sum of digits of

$$24x = 2 + 4 + x = 6+ x$$

Hence, $$6+x$$ is a multiple of $$3.$$

This is possible when $$6+x$$ is any one of these numbers $$0, \;3,\; 6,\; 9,$$ and so on ...

\begin{align}\text{For},&\\&6 + x = 0 \Rightarrow x = 0 - 6 = - 6\\\text{For},&\\&6 + x = 3 \Rightarrow x = 3 - 6 = - 3\\\text{For},&\\&6 + x = 6 \Rightarrow x = 6 - 6 = 0\\\text{For},&\\&6 + x= 9 \Rightarrow x = 9 - 6 = 3\\\text{For},&\\&6+x =12 \Rightarrow x=12-6=6\\&\dots \text{so on} \end{align}

Since $$x$$ is a single digit number, the sum of the digits can be $$6$$ or $$9$$ or $$12$$ or $$15$$ and thus, the value of comes to $$0$$ or $$3$$ or $$6$$ or $$9$$ respectively.

Thus, $$x$$ can have its value as any of the four different values $$0,\; 3,\; 6,\;$$ or $$9.$$

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