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# If 31z5 is a multiplier of $9$, where $z$ is a digit. What is the value of $z$?

Last updated date: 23rd Jul 2024
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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 $0\,or\,9$.