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Yesterday I downloaded a mobile app.Many of you may have also seen it.The working is as follows-

  1. Think of any two digit number from 10-99
  2. Sum up the digits of the number and subtract it from the original number.
  3. You get a number.Now compare this number with its corresponding image you are shown on the screen.See the image beside the number.Click next and magic!the phone exactly shows the image beside the number you had seen.

The screenshots of the number with the images are as follows- enter image description here

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So?How does it correctly guess the correct image each time?

Soham
  • 9,990

1 Answers1

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Answer is very interesting really!Loved the way the man who mad this has thought of....

At first take the number $10$ subtracting you get $9$.Then the the max value $99$.Subtracting you get $99-18=81.$.

So the range of possible values is between $9 \space to\space 81$.The numbers $0$ to $8$ and $82$ to $99$ is useless.

Second note an interesting this-

Carrying out the above said operation on $11,12,13,...,19$ you get the same value $9$.

From $20$ onwards upto $29$ you get $18$.

For $30-39$ you get $27$

For $40-49$ you get $36$ and so on....

Now,interestingly the possible values you get are multiples of $9$ from $9$ to $81$.(why not $99$?).ie.$9,18,27,36,45$ etc.

Now,we see that in the first page (also in all other pages) you have two multiples of $9$.So,how does the app know which is the correct answer?

Funny to say but all multiples of $9$ have the same images....

Nice app really!!

Hope my answer helps!!

Soham
  • 9,990
  • 2
    You have discovered the divisibility rule for 9. If you add up the digits of a number, you get a number that has the same remainder on division by $9$ as the original number. Subtracting the sum of the digits from a number always gives a multiple of $9$. This works because if your original number is $10a+b$, the sum of digits is $a+b$ and subtracting leaves $9a$. This works for any number of digits, not just two. – Ross Millikan Jan 30 '16 at 16:02