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

## 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)

## 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 \(9.\)