Solve for X

Number Theory Level 2

Given that $X>10000$ with $X \bmod 7 = 0$ , what is the smallest value of $X$ ?

The answer is 10003.

1 solution

Mathh Mathh
May 14, 2015

$10^4\equiv 3^4\equiv 9^2\equiv 2^2\equiv 4\equiv -3\pmod{7}$ , so $10^4+3=\boxed{10003}$ .

Moderator note:

Very simple and easy to understand. Bonus question: Given that $Y = 10^n +1$ for $n > 4$ with $Y \bmod 7 = 0$ and positive integer $n$ , what is the smallest value of $Y$ ?

Challenge Master: $10^4\equiv 10\cdot 10^3\equiv 3\cdot 10^3\equiv -3\pmod{7}$

$\stackrel{:3}\iff 10^3\equiv -1$ mod $7$ , so $\text{ord}_7(10)=6$ and $10^3\cdot 10^{\text{ord}_7(10)}\equiv 10^9\equiv -1$ mod $7$ with $9$ being the next least after $3$ to satisfy $10^k\equiv -1$ mod $7$ .

Or $10^n\equiv -1\,\Rightarrow 10^{n-3}\equiv 1$ mod $7$ and $n-3=\text{ord}_7(10)\iff n=9$ , since $n-3\neq 0$ .

mathh mathh - 6 years, 1 month ago

Solve for X

The answer is 10003.

1 solution

Moderator note:

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