# Thread: A question from Number Theory book of Thomas Koshy?

1. ## A question from Number Theory book of Thomas Koshy?

I have a question here from the book I have mentioned above

The question is - There are 4 nos between 100 and 1000, which are equal to the sum of the cubes of its digits. Three of them are 153, 371 and 407. Find the fourth no.

I still actually didn't get as to how to proceed to solve the problem. Can anyone explain the solution to me.

I found a pattern here - In each of the nos, the transition of the nos from one to the next, results in the change of only one digit

153
} 7
371
} 1
370
} 4
407

But I need a more plausible explanation as to how I can go about solving this and not use mere brute force!

2. Originally Posted by scw_0611
I have a question here from the book I have mentioned above

The question is - There are 4 nos between 100 and 1000, which are equal to the sum of the cubes of its digits. Three of them are 153, 371 and 407. Find the fourth no.

I still actually didn't get as to how to proceed to solve the problem. Can anyone explain the solution to me.

I found a pattern here - In each of the nos, the transition of the nos from one to the next, results in the change of only one digit

153
} 7
371
} 1
370
} 4
407

But I need a more plausible explanation as to how I can go about solving this and not use mere brute force!
I can't help you show there are 4 solutions, or even solve for the other 3, but I can show how to generate the last member if we are given one of the solutions.

Call the number "xyz" = 100x + 10y + z. Then you require that
$100x + 10y + z = x^3 + y^3 + z^3$

Or
$(x^3 - 100x) + (y^3 - 10y) + (z^3 - z) = 0$

$x(x^2 - 100) + y(y^2 - 10) + z(z^2 - 1) = 0$

Specifically what I'm after is the last term:
$x(x^2 - 100) + y(y^2 - 10) + z(z + 1)(z - 1) = 0$

If we have as a solution the number 371, note that $z(z + 1)(z - 1) = 0$ since z = 1. Thus we know that
$x(x^2 - 100) + y(y^2 - 10) = 0$
for this number.

There is another way to make this last term 0: set z = 0. So 370 should also be a solution.

-Dan

3. Thanks a lot. Got it.