Sum of digits of squares

**ADARSH** · Feb 7th 2009, 10:58 PM

Hello all,
Q. How many numbers between 1-2000 , have sum of digits of respective squares as 21 ?
I am interested to know the working rather than the answer, Thanks to all

**Opalg** · Feb 8th 2009, 06:30 AM

Originally Posted by ADARSH

Q. How many numbers between 1-2000 , have sum of digits of respective squares as 21 ?

The sum of the digits of a number is congruent to the number itself mod 9. So the sum of the digits of $n^2$ is congruent to $n^2$ mod 9. But a square has to be congruent to 0, 1, 4 or 7 (mod 9), and 21 is congruent to 3 (mod 9).

Conclusion: There are no numbers satisfying that condition.

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