On dividing a number by $19$, the difference between quotient and remainder is $9$. The number is

### Concept & Strategy When a problem provides an absolute difference between two division components but lacks sufficient equations to solve algebraically, the most efficient method is to reverse-engineer the provided multiple-choice options. $$ \text{Number} = (19 \times \text{Quotient}) + \text{Remainder} $$ ### Step-by-Step Solution * **Given:** * Divisor = $19$ * $| \text{Quotient} - \text{Remainder} | = 9$ * **Calculation:** * Test Option (a) $352$: $352 / 19 = 18$ with a remainder of $10$. $|18 - 10| = 8$. (Incorrect) * Test Option (b) $361$: $361 / 19 = 19$ with a remainder of $0$. $|19 - 0| = 19$. (Incorrect) * Test Option (c) $370$: $370 / 19 = 19$ with a remainder of $9$. $|19 - 9| = 10$. (Incorrect) * Test Option (d) $371$: $371 / 19 = 19$ with a remainder of $10$. $|19 - 10| = 9$. (Correct) ### Exam Strategy & Shortcut Instead of full division, use estimation. $19 \times 20 = 380$. All options are slightly below $380$, so the quotient for all of them must be either $18$ or $19$. For a quotient of $19$, the number is near $361$. If $Q = 19$, the remainder $R$ must be $10$ (to make the difference $9$). $361 + 10 = 371$. This matches option (d) instantly. ### Common Pitfall Assuming "difference between quotient and remainder" strictly means $Q - R = 9$. The phrasing implies an absolute difference $|Q - R| = 9$, meaning it could also be $R - Q = 9$. Staying flexible and checking the options prevents getting stuck on a single assumed equation. ### Final Answer Therefore, the correct answer is **371**.