A question mark.

Think of a number between 51 and 99, both included. Do the following operations:

  • Sum 75 to this number
  • Take out the first digit of the result
  • Add the result number to the digit that you take out in the step above
  • Subtract the number that you thought by the result of the last step.

The number that you get after these operations was 24, was not it? Can you explain the trick?

Explanation

Let xx be the number that you thought.

Then,

51x99(1)51 \leq x \leq 99\;\;\;\left(1\right)
51+75x+7599+7551+75 \leq x+75 \leq 99+75
126x+75174126 \leq x+75 \leq 174
126100x+75100174100126-100 \leq x+75-100 \leq 174-100
26x257426 \leq x-25 \leq 74
26+1x25+174+126+1 \leq x-25+1 \leq 74+1
27x2475(2)27 \leq x-24 \leq 75\;\;\;\left(2\right)

Subtracting (2) from (1) we get:

5127x(x24)997551-27 \leq x-\left(x-24\right) \leq 99-75
24xx+242424 \leq x-x+24 \leq 24
24242424 \leq 24 \leq 24

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