A 4-digit number N is such that when divided by 3, 5, 6, 9 leaves a remainder 1, 3, 4, 7 respectively. What is the smallest value of N?

To solve this, we first look at the difference between the divisors and their respective remainders.

3 minus 1 equals 2 5 minus 3 equals 2 6 minus 4 equals 2 9 minus 7 equals 2

Since the difference is a constant 2 in all cases, the number N follows the format: N equals LCM of 3, 5, 6, and 9 minus 2.

The LCM of 3, 5, 6, and 9 is 90. Therefore, the general form of the number is 90k minus 2, where k is an integer.

To find the smallest 4-digit number, we need 90k to be slightly greater than 1000. If we take k as 12, 90 times 12 equals 1080. N equals 1080 minus 2, which equals 1078.

Verification: 1078 divided by 3 leaves remainder 1. 1078 divided by 5 leaves remainder 3. 1078 divided by 6 leaves remainder 4. 1078 divided by 9 leaves remainder 7.

The smallest value of N is 1078. Thus, C is the correct option.