N is the largest 3-digit number, which when divided by 3, 4 and 6 leaves the remainder 1, 2 and 4 respectively. What is the remainder when N is divided by 7?

Step 1: We need to find a number that leaves a remainder of 1 when divided by 3, a remainder of 2 when divided by 4, and a remainder of 4 when divided by 6. This means we are looking for a number that fits the form 3a+1=4b+2=6c+4 for some integers a, b, and c.

Step 2: Simplify the equations to 3a=4b+1=6c. This means we are looking for a number that is one more than a multiple of 3, 4, and 6.

Step 3: The least common multiple of 3, 4, and 6 is 12. So, we are looking for numbers of the form 12d+1, where d is an integer.

Step 4: The largest 3-digit number is 999. So, we need to find the largest number of the form 12d+1 that is less than or equal to 999.

Step 5: Divide 999 by 12 to get 83 remainder 3. So, the largest number of the form 12d+1 that is less than or equal to 999 is 12*83+1=997.

Step 6: To find the remainder when 997 is divided by 7, divide 997 by 7 to get 142 remainder 5.

So, the remainder when N is divided by 7 is 5.