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Our good friend and numerologist, Professor Euclide Paracelso Bombasto Umbugio, has been busily engaged in testing on his desk calculator the 81x109 possible solutions to the problem of reconstrucing the following exact long division in which the digits were indiscriminately replaced by X save in the quotient where they were almost entirely omitted:

post-27517-1275577927131.png

Deflate the Professor! That is, reduce the possibilities to (81x109)0.

Because any number raised to the power of zero is one, the reader's task is to discover the unique reconstruction of the problem. The 8 is in the correct position above the line, making it the third digit of a five-digit answer. The problem is easier than it looks, yielding readily to a few elementary insights.

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Posted · Report post

Not sure what you meant by "5 digit answer", but here is one possible solution:

124 x 889 = 10030316

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Posted · Report post

5 digit number is 80809

The divided by number is 124

The number being divided is 10020316, last digit can be 6-9

10020316

-------- = 80809

124

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Posted · Report post

Long division takes me back to my school days.

When we have to drop down a second digit to continue the division, that gives us a zero in the answer. I started with this and came up with the following answer:

10020316 / 124 = 80809

:):):):)

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Posted (edited) · Report post

later...

Edited by openerbat
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Posted · Report post

18070991 / 199 = 90809

Donjar's and a few other's reply is also correct. And so is mine, I think. Since the puzzle states to find an unique answer, not sure if that's correct?

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Posted · Report post

Openerbat, unfortunately your answer, while mathematically correct, cannot be the unique solution for this problem.

199x9=1791, but the problem only asks for a three-digit answer.

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Posted · Report post

kaaadu and donjar got it right

10020316 / 124 = 80809

Only one solution. Also, don't forget to add the zeroes for the places not used, alexandra. It's a 5-digit quotient like the problem said.

Openerbat, if you work out the division for your solution you will see you have extra digits. If it says XXX that means only three digits can be there.

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Posted · Report post

There is only one answer. This is one of the best quick math puzzles I have ever seen. Thanks for making my evening.

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Posted (edited) · Report post

Only one solution for the divisor is possible, here is why,

The lonesome 8 tells us,

1) that some multiple of 8 when subtracted from a four digit number gives a two digit number.

2) That 8 times the mystery divisor gives some three digit number.

3) And that the difference from 1) is smaller than or equal to the first 2 digits of the division.

The numbers for 1) will be of this form:

10XX - 9XX = XX.

Since the largest three digit number is 999, which means the four digit number you're subtracting from can at most be 1098 (i.e. 1098 - 999 = 99, the largest two digit number). The smallest 4 digit number being 1000 allows that the smallest three digit number subtracted to be 901, so that the difference is a 2 digit number (the number 99).

The number has to be between 901 and 999, or to put another way, the subtracted number is bounded by 901 and 999.

2) stipulates that the 3 digit number be some multiple of 8.

What are then the multiples of 8 between 901, and 999?

They are, 904, 912, 920, 928, 936, 944, 952, 960, 968, 976, 984, and 992.

We also note that these are the multiples of 8 corresponding to the following multiplications,

113*8=904, 114*8=912,...,124*8=992.

So now the mystery divisor HAS to be between 113 and 124.

From 3) that means that the remainder can only be 10, 11, or 12; because it is a proper two digit number.

Entering 10, 11, and 12 into 1), means that the multiple of 8 from 2) is between 989, and 999.

So that 1009 - 10 = 999, or 1001 - 12 = 989.

The only multiple of 8 between 989 and 999 is 992, corresponding to the multiplication 8*124.

So the mystery division can only be 124.

hmm, I just realized I hadn't figured out the dividend yet...

Edited by marsupialsoup
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Posted (edited) · Report post

Here is to solve for the dividend.

The last 2 digits of the quotient has be start with a zero, or of the form 0X, otherwise the remainder from before could be divided by 124 when the 2nd last digit of the dividend is brought down. The only 4 digit multiple of 124, when we are restricted to multiplying by numbers less than 10, is 124*9=1116. The remainder from before is then 11, and the last two digits of the dividend is 16.

Adding 11 to 992 gives the remainder for the first division done with the first two digits of the quotient (the ones before the lonesome 8). That remainder is 1003. So the dividend looks now like XXXX0316.

The first four digits of the dividend, considered as a number, must give a difference of 10 when subtracted by some multiple of 124. Again because it is a four digit number subtracted by a three digit number to give a two digit number, it has to be of the form 10XX - 9XX = 10. The multiples of 124 in question now is narrowed down to 8 only, 124*8 = 992. Add 10 to 992 gives the first four digits of the dividend, 10020316.

It has to be that 10 020 316 / 124 = 80 809.

It gives the same answer as the other posters, I just wanted to make sure.

Edited by marsupialsoup
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