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# help

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The 5-digit number 5228X is a multiple of 6.  What digit is represented by X?

Jun 13, 2020

#1
+344
-1

5+2+2+8 = 17 so x can be 1 or 8 (if you want to know if a big number is divisable by a small number than you can add all the digits together but it does not always work)

Jun 13, 2020
#3
+549
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it obviously can't be 1 it's even!

amazingxin777  Jun 13, 2020
#2
+2

In order for a number to be divisible by 6, it needs to be divisible by both 3 and 2.

The divisiblity rule for 2 is if the end digit is an even number.

The divisiblity rule for 3 is if all the digits added up together can be divided by 3.

Even numbers are easy to spot, so let's focus on divisibility by 3.

5+2+2+8+X = 3n

17+X=3n

Now, we can use trial and error.

Case 1: n=6

17+X=18

X=1, which is not an even number, so discard this solution.

Case 1: n=7

17+X=21

X=4, which is an even number, so this is a valid solution.

At this point, we don't have to test for n=8 because the question states that there is only one digit for X.

Jun 13, 2020
#4
+8342
0

That means 5228X is divisible by 2 and 3.

So X must be an even number, and \(5 + 2 + 2 + 8 + X\) is divisible by 3.

But that means \(2 + 2 + 2 + 2 + X \) is divisible by 3. Taking mod 3 again on both sides,

\(2 + X\) is divisible by 3, and don't forget that X is an even digit.

\(X = 4\) is the only possible solution.

Jun 13, 2020