Is there any way we can find all the values of x in O(log) or O(1) time complexity.

X + X%5 = 1000

The brute force approach is thinking and putting different values of X and checking with RHS. But the things is I can't think of any other approach other than this for finding the value of X. Is there any other way we can actually find the value of X?


One solution is $X = 1000$. To find another solution, you only need to try values of $X$ in the vicinity of $1000$. For example, $\{999,998,997,996\}$. None of these satisfy the equation, so the solution is $X = 1000$. If the equation were $X - X\%5 = 1000$, then there would be solutions $X = \{1000,1001,1002,1003,1004\}$.

Extending this, the solution to $$X + X\%5 = 10^{18},$$ would be $X = 10^{18}$, and the solution to $$X-X\%5 = 10^{18}$$ would be $X = \{10^{18},1+10^{18},\dots,10^{18}+4\}$.

  • $\begingroup$ Thanks a lot. That does seem the right approach. $\endgroup$ – arqam Jun 25 '19 at 17:17

Let's write $X_5$ for $X\%5$, i.e., $X_5$ is the number such that $X_5 \equiv X \pmod 5$ and $0 \le X_5 < 5$; we must solve $X + X_5 = 1000$. Note that $(X_5)_5 = X_5$. Now, reducing modulo $5$, $$X_5 + X_5 \equiv 0 \pmod 5$$ so $X_5 = 0$. Therefore, since $9996 < X \le 1000$, $X$ = 1000.

This approach avoids any brute-force checking.


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