Correct Option

The problem requires finding the smallest 4-digit number N that satisfies a set of remainder conditions:

N divided by 3 leaves a remainder of 1 (N ≡ 1 (mod 3))
N divided by 5 leaves a remainder of 3 (N ≡ 3 (mod 5))
N divided by 6 leaves a remainder of 4 (N ≡ 4 (mod 6))
N divided by 9 leaves a remainder of 7 (N ≡ 7 (mod 9))

Observe the difference between each divisor and its corresponding remainder:

3 − 1 = 2
5 − 3 = 2
6 − 4 = 2
9 − 7 = 2

Since this difference is constant (2), it implies that N + 2 is perfectly divisible by 3, 5, 6, and 9. Therefore, N + 2 must be a multiple of the Least Common Multiple (LCM) of these divisors.

Calculate the LCM of 3, 5, 6, and 9:

Prime factorization of 3 = 3¹
Prime factorization of 5 = 5¹
Prime factorization of 6 = 2¹ × 3¹
Prime factorization of 9 = 3²

The LCM is obtained by taking the highest power of each prime factor present: LCM(3, 5, 6, 9) = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90.

Thus, N + 2 must be a multiple of 90. We can write this as N + 2 = 90k, where k is a positive integer. Rearranging for N, we get N = 90k - 2.

To find the smallest 4-digit number N, we set N ≥ 1000:

90k - 2 ≥ 1000

90k ≥ 1002

k ≥ 1002 / 90

k ≥ 11.133...

Since k must be an integer, the smallest integer value for k that satisfies the condition is 12.

Substitute k = 12 into the equation for N:

N = 90 × 12 - 2

N = 1080 - 2

N = 1078

The number 1078 is a 4-digit number and satisfies all the given remainder conditions.

Incorrect Options

The following options do not satisfy all the given remainder conditions:

Option (1) 1068:
- 1068 ÷ 3 leaves a remainder of 0 (expected 1).
- 1068 ÷ 6 leaves a remainder of 0 (expected 4).
- 1068 ÷ 9 leaves a remainder of 6 (expected 7).
Therefore, 1068 is not the correct number.
Option (2) 1072:
- 1072 ÷ 5 leaves a remainder of 2 (expected 3).
- 1072 ÷ 9 leaves a remainder of 1 (expected 7).
Therefore, 1072 is not the correct number.
Option (4) 1082:
- 1082 ÷ 3 leaves a remainder of 2 (expected 1).
- 1082 ÷ 5 leaves a remainder of 2 (expected 3).
- 1082 ÷ 6 leaves a remainder of 2 (expected 4).
- 1082 ÷ 9 leaves a remainder of 2 (expected 7).
Therefore, 1082 is not the correct number.