7. Let $n$ be a positive integer greater than 1 . If $2 n$ is divided by 3 , the remainder is 2 . If $3 n$ is divided by 4 , the remainder is 3 . If $4 n$ is divided by 5 , the remainder is 4 . If $5 n$ is divided by 6 , the remainder is 5 . What is the least possible value of $n$ ?
Answer: 61
Solution: We have $2 n \equiv 2(\bmod 3), 3 n \equiv 3(\bmod 4), 4 n \equiv 4(\bmod 5)$, and $5 n \equiv 5(\bmod 6)$. This is equivalent to saying that $n \equiv 1 \bmod 3,4,5$, and 6 . The smallest such $n$ is one more than the LCM of 3,4 , 5 , and 6 , which is 61 .