On dividing a certain binary number exactly (remainder = 0) by 112, the…

For a binary division that is exact (remainder = 0), the identity Dividend = Divisor × Quotient holds with no remainder term. So, given the divisor and the required exact quotient, the missing dividend is found by multiplying the divisor by the quotient, working entirely in binary.

1. Divisor = 112 (decimal 3); required quotient = 102 (decimal 2); remainder = 0.

2. Dividend = Divisor × Quotient = 112 × 102.

3. Multiplying a binary number by 102 is a left-shift by one bit: 112 shifted left once gives 1102.

4. So the dividend (the missing binary number) is 1102.

Cross-check by dividing 1102 back by 112 (binary long division): the quotient is 102 with remainder 0 — an exact match. Note that 1112 (decimal 7) divided by 112 also gives a quotient of 102, but leaves a remainder of 1 — that division is not exact, so it does not satisfy the clean, remainder-free relationship that 1102 does.

So the required binary number is 110.