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A Number TrickSelect a three digit number in which the first and the last digit differ by at least two. Construct a second number by reversing the order of digits in the first. Form a third number by taking the difference of the first two. Reverse the order of digits in the third number to construct a fourth, and add the third and fourth number. The result is the number 1089. Example: Select the number 732. Note: 7 - 2 = 5 > 2, making this number a valid choice. Next, construct the number 237 by reversing the order of digits. Take the difference: 732
- 237 = 495
From 495, construct the number 594, and add: 495
+ 594 = 1089.
Proof: Let the originally selected number be represented by where a, b, c, are integers between 0 and 9 inclusive, and occupy the hundreds, tens, and units places, respectively. We may assume, without loss of generality, that a > c. Now, construct a new number by reversing the order of digits
Take the difference
and simplify to
where
Since
2
where
(10
. 9
![]() ![]()
If the
integer 100
![]() ![]() is the
number sought. Let us find all possible values of
Thus,
and the sum of these last two numbers is
But,
for all values of a considered, ( |
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