Number Theory

Question

+1

34

1

+2729

Find the number of solutions to
N &\equiv 2 \pmod{6}, \\
N &\equiv 2 \pmod{7}, \\
N &\equiv 2 \pmod{8}
in the interval 0 \le N < 1000.

LiIIiam0216 Jul 17, 2024

NotThatSmart · Answer 1 · 2024-07-17T02:30:31

For N to satisfy all the choices given, we must find the LCM of 6, 7, and 8.

We have that $LCM[6,7,8]=168$

Since they are all containing remainder 2, we have the equation

$168x+2 \leq 1000$

Since we want the max number, we set the equations to equal each other and solve for x. We get

$168x+2=1000\\ 168x=998\\ x=\frac{499}{84} \approx 5.940$

We round down, so our final answer is 5.

Thanks! :)