Question

If 31z5 is a multiplier of \[9\], where \[z\] is a digit. What is the value of \[z\]?

Answer 1

Hint: We apply the divisibility formula theorem of \[9\]. We find the sum of the digits of \[31z5\], which will be divisible by \[9\]. We try to make an equation after finding the multiple of \[9\]. By solving the linear equation, we find out the possible value of \[z\].

Complete step-by-step solution:
We are going to use the divisibility formula theorem of \[9\]. We know that a number is divisible by \[9\] only when the sum of the digits of the number is divisible by \[9\].
For example, we take a number (abc). Here hundredth, tenth, unit placed digits are a, b, c respectively.
So, (abc) will be divisible by \[9\], only when (a+b+c) is divisible by \[9\].
We take \[4756\]. The sum of the digits \[4 + 7 + 5 + 6 = 22\], which is not divisible by \[9\]. So, \[4756\] is not divisible by \[9\].
For our given problem, \[31z5\] is a multiple of \[9\]. This means \[31z5\] is divisible by \[9\].
So, the sum of the digits \[31z5\] has to be divisible by \[9\].
This means \[3 + 1 + z + 5 = 9\,or\,18\]
 Here, \[z\]is a single digit. Now, we find the multiple of \[9\].
\[
   \Rightarrow z = 9 - 9 = 0 \\
   \Rightarrow z = 18 - 9 = 9 \]
Therefore, there are two possible answers that are \[0\,and\,9\].

Note: We need to keep in mind that \[z\] is only a single digit. So, since \[z\] could be any single digit the answer could be \[0\,or\,9\]. Since the numbers divisible, which is \[9\,\,and\,\,18\], could be obtained by adding the numbers and \[z\] coming out to be \[0\,or\,9\].