Mystery Number Puzzle

[Otis]

Find a five-digit number containing no zeros and no repeated digits that satisfies the following conditions: the last digit is the sum of the first and second digits; the fourth digit is one less than the last digit; the second digit is four more than the first digit; and the sum of all of the digits is 24.

 

[lev888]

Find a five-digit number containing no zeros and no repeated digits that satisfies the following conditions: the last digit is the sum of the first and second digits; the fourth digit is one less than the last digit; the second digit is four more than the first digit; and the sum of all of the digits is 24.

26178

 

[lookagain]

I have the actual work in the spoiler to support the answer, including that it shows
that there is a unique solution.

Spoiler

Let the listing of digits of the number from the left to the right be:
a, b, c, d, e.

a + b = e
d = e - 1
b = a + 4
a + b + c + d + e = 24
----------------------------------

d = e - 1 ---> e = d + 1
So, a + b = d + 1 --->
d = a + b - 1
Recall e = a + b.

Substitute what d & e equal into the fourth equation at the top:

a + b + c + (a + b - 1) + (a + b) = 24 --->
3a + 3b + c = 25 *

b = a + 4 ---> b - a = 4 ---> 3b - 3a = 12 **

Add equations * and ** together:

6b + c = 37 ---> c = 37 - 6b.

Potential possible choices for b (so c can be legit) are: 5 or 6. Then start
back-substituting for the other possible digits.

b = 5, c = 7, a = 1, d = 7 . . . . Stop. There is a repeated digit in this route.
b = 6, c = 1, a = 2, d = 7, e = 8 . . . . . There are no repeated digits here.

This work shows that there are no other possible answers.

The 5-digit number is \(\displaystyle \ \)26,178.