Math Classroom 20161120: 297 and Magic 1089

Question: How many 3-digit numbers possess the following property: after subtracting 297 from such a number, we get a 3-digit number consisting of the same digits in the reverse order?

Answer: lets put a as the digit in ones, b as the digit in tenth, c as the digit in hundredth, therefore, the 3-digit number is:

100a + 10b + c

the reversed 3-digit number is:

100c + 10b + a

Then the equation can be built:

Step 1) 100a + 10b + c – 297 = 100c + 10b + a
Step 2) 100a + 10b + c – (100c + 10b + a) = 297
Step 3) 100a + 10b + c  – 100c – 10b – a = 297
Step 4) 99a – 99c = 297
Step 5) 99(a – c) = 297
Step 6) a – c = 3

We know the difference between the first digit and the last digit is 3, then the all possible combination are:

                  Hundredth                  Tenth                    Ones
Option 1:     3                                 0 to 9                       0
Option 2:     4                                 0 to 9                       1
Option 3:     5                                 0 to 9                       2
Option 4:     6                                 0 to 9                       3
Option 5:     7                                 0 to 9                       4
Option 6:     8                                 0 to 9                       5
Option 7:     9                                 0 to 9                       6

the first option does not work since the reversed number is not a 3-digit number.

The answer should be in Option 2 to Option 7; therefore, 10*6 = 60 is the amount of 3-digit numbers we have.

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Here is some material you want to read about Magic 1089

Magic 1089

Here’s a cool mathematical magic trick. Write down a three-digit number whose digits are decreasing. Then reverse the digits to create a new number, and subtract this number from the original number. With the resulting number, add it to the reverse of itself. The number you will get is 1089!

For example, if you start with 532 (three digits, decreasing order), then the reverse is 235. Subtract 532-235 to get 297. Now add 297 and its reverse 792, and you will get 1089!

Presentation Suggestions:
You might ask your students to see if they can explain this magic trick using a little algebra.

The Math Behind the Fact:
If we let a, b, c denote the three digits of the original number, then the three-digit number is 100a+10b+c. The reverse is 100c+10b+a. Subtract: (100a+10b+c)-(100c+10b+a) to get 99(a-c). Since the digits were decreasing, (a-c) is at least 2 and no greater than 9, so the result must be one of 198, 297, 396, 495, 594, 693, 792, or 891. When you add any one of those numbers to the reverse of itself, you get 1089!

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