### Instruction: What is the largest number $k$ less than 1000 such that the remainder is 1 when $k$ is divided by each of 3, 4, 5, 6, and 7?### Response: I need to find a number $k$ that is congruent to 1 modulo 3, 4, 5, 6, and 7.
This means that $k$ is one more than a common multiple of 3, 4, 5, 6, and 7.
The least common multiple of these numbers is 420, so the smallest possible value of $k$ is 420 + 1 = 421.
To find the largest possible value of $k$ less than 1000, I can add multiples of 420 to 421 until I get close to 1000.
The largest multiple of 420 that is less than 1000 is 840, so the largest possible value of $k$ is 840 + 1 = 841.