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

• We need the smallest number N such that 7 × N is a repunit (a number made up entirely of the digit 1, e.g. 111, 1111, 11111, 111111, …).

• This is equivalent to finding the smallest repunit that is exactly divisible by 7, since N = (repunit) ÷ 7.

• Testing repunits in order: 1 ÷ 7, 11 ÷ 7, 111 ÷ 7, 1111 ÷ 7, 11111 ÷ 7 all leave remainders, so none of them is divisible by 7.

• The next repunit, 111111, gives 111111 ÷ 7 = 15873 exactly, with no remainder.

• Verifying the multiplication digit by digit: 3×7 = 21 (write 1, carry 2); 7×7 + 2 = 51 (write 1, carry 5); 8×7 + 5 = 61 (write 1, carry 6); 5×7 + 6 = 41 (write 1, carry 4); 1×7 + 4 = 11 (write 11), yielding 111111.

• Since 111111 is the smallest repunit divisible by 7, 15873 is the smallest number whose product with 7 consists entirely of ones, matching option (d).