Same Unit Digit In Different Bases

When a number is written in base $n$ , the units digit of the number is the same as the remainder when the number is divided by $n$ . So, basically, the problem wants us to find a number $x$ which leaves a remainder of 3 when divided by 4, 5, 6, 7 or 8.

$x \equiv 3 \pmod {4, 5, 6, 7, 8}$

Since we know that if $x \equiv n \pmod{a}$ and $x \equiv n \pmod{b}$ , then $x \equiv n \pmod{lcm(a,b)}$ ,

$x \equiv 3 \pmod {lcm(4, 5, 6, 7, 8)}$

$x \equiv 3 \pmod {840}$

This means $x$ is of the form $840k+3$ for nonnegative integers $k$

Since $x = 3$ when $k = 0$ , then the next smallest integer which satisfies the requirement is $\boxed{843}$ , when $k = 1$

You can Use the python code for solving the problem: