Answer

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**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 \[0\,or\,9\].

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