+1286 Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{7}.
+1926 We need to solve the system of equations
\(N \equiv 2 \pmod{5}, \\ N \equiv 2 \pmod{7}\)
Well, let's analyze this equation.
N leaves a remainder of 2 when divided by 5 and 7.
This means that N is two greater than the LCM of 7 and 5.
This is because since it leaves the same remainder for both numbers, then it must be the same number.
Therefore it is the LCM we are looking for.
We have
\(LCM[5, 7] = 35\)
Adding two, we have \(35+2 = 37\)
thus, our answer is 37.
Thanks! :)
