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$N$ is a 6 digit Natural number such that its sum of the digits is $43$.

Find $N$ if Exactly One of the statements below is False:

$(1)$ $N$ is a Perfect Square

$(2)$ $N$ is a Perfect Cube

$(3)$ N $\lt 500000$

My Try: The least 6 digit number with Sum of the digits $43$ is $169999$ and the Highest number is $999970$.

Case(i):-> Let Statement 2 is False, That is $N$ is not Perfect Cube.

Now since $N$ $\lt 500000$ and $N$ Last digit can only be either of 0,1,4,5,6 and 9.

Since Least number with sum of digits $43$ is $169999$ and Highest number $\lt 500000$ with sum of digits $43$ is $499993$. Now we need to check a Perfect Square in the Interval $[169999\: 499993]$

I tried to list out numbers in the above interval , but its becoming very Lengthy...Help Required. Thanks

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  • $\begingroup$ Just to begin, if it is a perfect square and a perfect cube, it has be to a sixth power. But working modulo $9$, and using $\varphi(9) = 6$ you see that you cannot obtain the sum of the digits to be $43$. So one of the first two conditions has to fail. $\endgroup$ – Andreas Caranti Aug 31 '14 at 10:58
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First of all, statement $2$ is false :

Since the sum of the digits of $N$ is $43\equiv 7\pmod9$, we know that $N\equiv 7\pmod9$. However, since there is no $m\in\mathbb N$ such that $m^3\equiv 7\pmod9$, we know that statement $2$ is false.

As a result, we have to find $N$ which satisfies both statement $1$ and $3$. Here, note that $$m^2\equiv 7\pmod9\iff m\equiv 4,5\pmod9.$$ This fact will enable you to eliminate many cases.

So, we have $66$ cases to check if the sum of the digits of $m^2$ is $43$ : $$m=418+9t, 419+9t\ \ (t=0,1,\cdots,32).$$

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I used a computer and find out the following. In the set of $6$-digit numbers $N$ with sum of digits $43$ there are

  • $5$ perfect squares: $499849$, $678976$, $693889$, $784996$, $786769$
  • $0$ perfect cubes
  • $4317$ numbers smaller than $500000$

Therefore the wanted $N$ cannot be a perfect cub. Since there is only one perfect square smaller than $500000$, your number is $N = 499849$.

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  • $\begingroup$ Is using computer is fair in this question? I mean, since he uses the tag number-theory, I expected a non-computer answer. $\endgroup$ – Realdeo Aug 31 '14 at 10:50
  • $\begingroup$ I do not know. I don't know any appropriate number-theoretic theorems/approaches. I would be able to count numbers $N$ which are smaller than $500000$, but I find (number) of those from the first and the second condition too hard to compute in a smart way. $\endgroup$ – Antoine Aug 31 '14 at 10:55

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