SOLUTION: If 31z5 is a multiple of 9, where z is a digit, what is the value of z?

Question 192950: If 31z5 is a multiple of 9, where z is a digit, what is the value of z?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If 31z5 is a multiple of 9 (where z is a single digit), then the digits MUST add to a number that is a multiple of 9 (hopefully, they will add to 9 itself).

So the digits must add to 0, 9, 18, 27, 36, 45, 54, 63, 72, etc...

So this gives us the equation:

where "k" is an integer (ie whole number)

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So if (the smallest possible k value where we get a solution), then...

Solve for "z" to get: --->

So when , then giving the number 3105

So 3105 is divisible by 9. Check: remainder 0.

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If , then...

Solve for "z" to get: --->

So when , then giving the number 3195

So 3195 is divisible by 9. Check: remainder 0.

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If , then...

Solve for "z" to get: --->

But since "z" is a single digit, this means that . So ANY value of "k" greater than 2 will give us a "z" value greater than 9. So this means that we're done looking for values of "z".

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Answer:

So the two values of "z" are 0 and 9 produce the multiples of 9 3105 and 3195 respectively.
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