Problem Solving Question

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October 1st 2010, 10:17 AM
matgrl

Problem Solving Question

Does anyone know what the five digit number 5DDDD is when it is divisible by 6.

What is the digit D?

I have this so far...

50000/6 = 8333.33 remaining

51111/6 = 8518.5

52222/6 = 8703.66 remaining

53333/6 = 8888.83 remaining

54444/6 = 9074

55555/6 = 9259.166 remaining

56666/6 = 94444.33 remaining

Does this mean that 4 is the only digit of D that is divisible by 5DDD, because it is the only digits that goes into this problem evenly?

I honestly do not know if this is right...these are just my ideas. Any help or ideas would be greatly appreciated.
October 1st 2010, 10:41 AM
undefined

Quote:

Originally Posted by matgrl

Does this mean that 4 is the only digit of D that is divisible by 5DDD, because it is the only digits that goes into this problem evenly?

Exhaustive search is a valid method, but if you want to do that you will need to check 57777 through 59999 as well. (And indeed 54444 is the only solution.)
October 1st 2010, 12:56 PM
Soroban

Hello, matgrl!

Quote:

$\text{A five-digit number of the form }N = 5DDDD\text{ is divisible by 6.}$
$\text{What is the digit }D?$

As undefined pointed out, an exhaustive search is a valid approach.

We can use some knowledge to aid in our search.

Since $N$ is divisible by 6, it is divisible by 2 and 3.

Since $N$ is divisible by 2, it must be an even number.
. . An even number ends in an even digit: 0, 2, 4, 6, or 8.

Since $N$ is divisible by 3, the sum of its digits must be a multiple of 3.

. . $\begin{array}{ccc}D & N & \text{Digital sum} \\ \hline<br /> 0 & 50000 & 5 \\ 2 & 52222 & 13 \\ 4 & 54444 & 21 \\ 6 & 56666 & 29 \\ 8 & 58888 & 37 \end{array}$

And we see that only $54444$ is divisible by 6.
October 4th 2010, 05:24 AM
matgrl

I understand your chart and I think this is a great idea, but I have one question. How does54444 show that it is divisible by 6? The digitial sum is not an even number. What exactly does this digital sum show us?
October 4th 2010, 05:27 AM
undefined

Quote:

Originally Posted by matgrl

I understand your chart and I think this is a great idea, but I have one question. How does54444 show that it is divisible by 6? The digitial sum is not an even number. What exactly does this digital sum show us?

A positive integer is divisible by 3 if and only if its digital sum is divisible by 3. So you can actually repeat the digitial sum process iteratively (take the digital sum of the digital sum of the digital sum....) until you reach a single digit, if you wanted to.

An integer is divisible by 6 if and only if it is divisible by 2 and it is divisible by 3.