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Hi. I'd like to know how many 6 digit numbers are there of a combination of numbers 0, 2, 3, 5, 6, and 9 which can be divided to 5. None of the 6 digit numbers should be the same. But its allowed to repeat those 6 numbers in a 6 digit number. Thanks in advance. Your early response is appreciated.

Lilia

It sounds like you want to arrange the digits 0, 2, 3, 5, 6, and 9 in any order to get a six digit number that is divisible by 5. Since there are 6 numbers, the number of ways to arrange them is 6!, but then we have to throw some out.

To start with, to be divisible by 5, they need to end in 0 or 5. That means we can only keep

2/3 of the numbers. Now 6! = 6*5*4*3*2*1, of which I can drop the last 1, giving 6*5*4*3*2.

It is not that difficult to multiply them out, but it is even easier to adjust the list first.

Since the string of numbers needs to be multiplied by 2/3,

note that (2/3)3 = 2, so that changes the 3 into a 2, giving 6*5*4*2*2.

Now of those numbers, 1/6 of them start with 0, and they can be dropped.

This means we only have 5/6 of the numbers left, and (5/6)6 = 5.

That makes the leading 6 into a 5, so now we have 5*5*4*2*2.

Since it is known that 5*2 is 10, and there are 2-5's at the start and 2-2's at the end,

that makes the number into 4*10*10 = 400.

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