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# The four-digit number is divisible by nine, with the tens digit and the units digit. How many such four-digit numbers

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The four-digit number \(25AB\) is divisible by nine, with \(A\) the tens digit and \(B\) the units digit. How many such four-digit numbers could \(25AB\) represent?

Oct 4, 2019

#1
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For AB I got 11, 02, 20, 29, 38, 47, 56, 65, 74, 83, and 92. I'm not sure if that's all of them.

Oct 4, 2019
#2
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25AB

The sum of the digits must be a multiple of 9......so.....

2 + 5  +  A  + B  =   9M

A + B   =  9M - 7

If  M =  1                    M  =  2                M  = 3

A + B  =  2                A + B  =  11         A + B  = 20

A , B  =                     A , B =                 No possibilities  for A, B when  M > 2

0, 2                           2, 9

and the reverse        3, 8

4, 7

5, 6

and the reverses

So....by my count.....we have 5(2) =  10 such numbers   Oct 4, 2019
#3
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If A,B  can be the same integers....you are correct, HSM  !!!

[ I assumed that they must be unique ]

Whatever.....our answers only differ by 1, so  it probably depends upon our assumptions  ....LOL!!!   CPhill  Oct 4, 2019
#4
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I don't think you counted the 11 case. 2511 would also work because 2+5+1+1=9

hellospeedmind  Oct 4, 2019
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hellospeedmind  Oct 4, 2019
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Yep...if A,B  can be the same integers, you are correct

[ I assumed that they had to be different ]   CPhill  Oct 4, 2019