SOLUTION: Only four integers between 100 and 1000 equal the sum of the cubes of their digits. Three of these are 153, 370, and 407. What is the fourth?

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Question 540914: Only four integers between 100 and 1000 equal the sum of the cubes of their digits. Three of these are 153, 370, and 407. What is the fourth?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Lucky for us, we got those 3 numbers.
The problem tells us that for 370, the sum of the cubes of the digits is
3%5E3%2B7%5E3%2B0%5E3=3%5E3%2B7%5E3=370
Then, for 371, the sum of the cubes of the digits must be
3%5E3%2B7%5E3%2B1%5E3=3%5E3%2B7%5E3%2B1=371
BINGO!
The fourth number is 371.
No, I did not realize immediately.
I found it through a lot of logical work
After I found it, I immediately realized that it was so obvious (in hindsight) that if one of the solutions ended in zero, the next integer would also be a solution.
That is the kind of question that would be easily answered by a 13-year old in the artofproblemsolving forum. (That's full of kids with at least semi-realistic hopes to get to the Math Olympiad).