When a certain number is multiplied by 7, the product entirely comprises ones only (1111...). What is the smallest such number?

The correct option is 15873.

To find the smallest number that, when multiplied by 7, yields a product consisting entirely of ones (1111...), we can perform long division. We divide a sequence of ones (1, 11, 111, 1111, ...) by 7 until we get a remainder of zero.

Let's perform the division step-by-step:

1. Divide 1 by 7: quotient is 0, remainder is 1.

2. Bring down the next 1 to make 11: 11 divided by 7 gives 1 with a remainder of 4.

3. Bring down the next 1 to make 41: 41 divided by 7 gives 5 with a remainder of 6.

4. Bring down the next 1 to make 61: 61 divided by 7 gives 8 with a remainder of 5.

5. Bring down the next 1 to make 51: 51 divided by 7 gives 7 with a remainder of 2.

6. Bring down the next 1 to make 21: 21 divided by 7 gives 3 with a remainder of 0.

Since the remainder is now 0, the division terminates completely.

$111111÷7=15873$

Therefore, the smallest such number is 15873.