Q:

A three-digit number has two properties

0

A three-digit number has two properties. The tens-digit and the ones-digit add up to 5. If the number is
written with the digits in the reverse order, and then subtracted from the original number, the result is 792. Use
a system of equations to find all of the three-digit numbers with these properties.

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Let a be the hundreds digit, b the tens digit, and c the ones digit. Then the first condition says that b + c = 5.
The original number is 100
a + 10b + c, while the reversed number is 100c + 10b + a. So the second condition is
792 = (100
a + 10b + c) - (100c + 10b + a) = 99a - 99c
So we arrive at the system of equations
b + c = 5
99
a - 99c = 792
Using equation operations, we arrive at the equivalent system
a - c = 8
b + c = 5
We can vary
c and obtain infinitely many solutions. However, c must be a digit, restricting us to ten values (0 -
9). Furthermore, if
c > 1, then the first equation forces a > 9, an impossibility. Setting c = 0, yields 850 as a
solution, and setting
c = 1 yields 941 as another solution.

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