किसी 3-अंकों की संख्या ABC को D से गुणा करने पर गुणनफल 37DD प्राप्त होता है, जहाँ A, B, C और D भिन्न शून्येतर अंक हैं । A + B + C का मान क्या है ?

Correct Answer: (A) 18

The product is $\text{ABC}\times D=37DD$$\text{ABC}$ $\times$ D = 37DD. Since ABC is a 3-digit number, $\text{ABC}\le 999$$\text{ABC}$ $\le$ 999, which implies $37DD / D$$\le$ 999. Testing valid digits $D$$\ge$ 4: For $D = 4$, $\frac{3744}{4}=936$$\frac{3744}{4}$ = 936. Here $\text{ABC}=936$$\text{ABC}$ = 936, giving distinct non-zero digits $A = 9, B = 3, C = 6$, and $D = 4$. For $D = 5$, $3755 / 5 = 751$ (repeats digit 5). For $D$$\in$ \{6, 7, 8, 9\}, $37DD$ is not divisible by $D$. Thus, $D = 4$ is the unique solution, and $A + B + C = 9 + 3 + 6 = 18$.