A 3-digit number ABC, on multiplication with D gives 37DD where A, B, C and D are different non-zero digits. What is the value of A + B + C ?
Correct Option: 18
We are given that a 3-digit number ABC multiplied by a single-digit D gives the 4-digit number 37DD, where A, B, C, D are distinct non-zero digits. Let's write this in equation form: ABC × D = 37DD. Notice that the number 37DD can be written as 3700 + 11 × D. Let's test possible values for D (non-zero digits 1 through 9): If D = 5: 3755 / 5 = 751. Here ABC = 751, D = 5. Let's check digits: A = 7, B = 5, C = 1, D = 5. But wait! C and D cannot be the same digit if digits are distinct ('where A, B, C and D are different non-zero digits'). Here C = 1, D = 5 (different). Let's check multiplication: 751 × 5 = 3755. Here the result ends in 55, which matches 37DD (where D=5, so 3755). The digits are A=7, B=5, C=1, D=5. Wait, B is 5 and D is 5, but the problem states 'A, B, C and D are different non-zero digits'! So D cannot be 5. Let's test other values of D: If D = 6: 3766 / 6 = 627.66 (not an integer). If D = 7: 3777 / 7 = 539. Let's check: ABC = 539, D = 7. Multiplication: 539 × 7 = 3773. Here the product ends in 73, but we need 37DD (where the last two digits are both D, i.e., 77). So 539 × 7 = 3773 does not match 3777. Let's test D = 9: 3799 / 9 = 422.11 (not an integer). Let's check all possibilities or use divisibility: 37DD is divisible by D. Also 37DD = 3700 + 11D = 37 × 100 + 11D = 339 × 11 + 1 + 11D... wait. Let's test D = 8: 3788 / 8 = 473. Let's check: ABC = 473, D = 8. Multiplication: 473 × 8 = 3784 (does not end in 88). Let's test D = 6? Wait, what about D = 9? Let's check ABC = 419, D = 9 => 419 × 9 = 3771. What about 37DD / D = ABC? Let's test D = 4: 3744 / 4 = 936. Let's check: ABC = 936, D = 4. Multiplication: 936 × 4 = 3744! Let's check the digits: A = 9, B = 3, C = 6, D = 4. Are all digits distinct and non-zero? A = 9, B = 3, C = 6, D = 4. Yes, all 4 digits (9, 3, 6, 4) are distinct and non-zero! We need to find the value of A + B + C: A + B + C = 9 + 3 + 6 = 18. Thus, the correct option is 18 (Option A).
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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 ?
सही उत्तर: 18
दिया गया है कि 3-अंकों की संख्या ABC को D से गुणा करने पर 37DD प्राप्त होता है, जहाँ A, B, C और D भिन्न शून्येतर अंक हैं। यदि हम D = 4 लें, तो 3744 को 4 से भाग देने पर 936 प्राप्त होता है। अर्थात् ABC = 936 और D = 4। गुणनफल जाँचने पर: 936 × 4 = 3744, जो कि 37DD के रूप (जहाँ D=4) से पूर्णतः मेल खाता है। यहाँ सभी अंक A = 9, B = 3, C = 6, और D = 4 भिन्न और शून्येतर हैं। अतः A + B + C = 9 + 3 + 6 = 18 होगा। सही विकल्प (a) है।