What is the smallest number greater than 1000 that when divided by any one of the numbers 6, 9, 12, 15, 18 leaves a remainder of 3?
The correct answer is (C) 1083.
Explanation:
First, we find the LCM of 6, 9, 12, 15, and 18. Prime factorizations: 6 = 2×3, 9 = 3^2, 12 = 2^2×3, 15 = 3×5, 18 = 2×3^2. LCM = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180. We need the smallest number greater than 1000 that leaves a remainder of 3 when divided by 180. Let k be an integer such that 180k + 3 > 1000. For k = 5, 180 × 5 + 3 = 903 (not greater than 1000). For k = 6, 180 × 6 + 3 = 1080 + 3 = 1083. Wait, let us check k=5: 903. For k=6: 1083. Is there a smaller multiple? Let's check 180×5 = 900, 180×6 = 1080. 1080 + 3 = 1083. Let us check option 1073: 1073 - 3 = 1070, is 1070 divisible by 180? No. Let's check 1083 - 3 = 1080, which is divisible by 180 (180 × 6 = 1080). Thus 1083 leaves remainder 3 and is greater than 1000.
हिंदी में प्रश्न एवं आदर्श उत्तर
1000 से बड़ी वह लघुतम संख्या कौन-सी है जिसे 6, 9, 12, 15, 18 में से किसी एक से भी विभाजित करें, तो शेषफल 3 बचे?
सही उत्तर (C) 1083 है।
व्याख्या:
First, we find the LCM of 6, 9, 12, 15, and 18. Prime factorizations: 6 = 2×3, 9 = 3^2, 12 = 2^2×3, 15 = 3×5, 18 = 2×3^2. LCM = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180. We need the smallest number greater than 1000 that leaves a remainder of 3 when divided by 180. Let k be an integer such that 180k + 3 > 1000. For k = 5, 180 × 5 + 3 = 903 (not greater than 1000). For k = 6, 180 × 6 + 3 = 1080 + 3 = 1083. Wait, let us check k=5: 903. For k=6: 1083. Is there a smaller multiple? Let's check 180×5 = 900, 180×6 = 1080. 1080 + 3 = 1083. Let us check option 1073: 1073 - 3 = 1070, is 1070 divisible by 180? No. Let's check 1083 - 3 = 1080, which is divisible by 180 (180 × 6 = 1080). Thus 1083 leaves remainder 3 and is greater than 1000.