गुणनफल 1 × 5 × 10 × 15 × 20 × 25 × 30 × 35 × 40 × 45 × 50 × 55 × 60 के अंत में कितने शून्य होंगे?
सही उत्तर: (d) 15
शून्यों की संख्या ज्ञात करने के लिए हमें अभाज्य गुणनखंडन में 5 की कुल घातों की गणना करनी होगी। दी गई श्रृंखला में 5 के गुणज 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 हैं (कुल 12 गुणज)। इनमें से 25 में 5 की दो घातें हैं तथा 50 में भी 5 की दो घातें हैं। इस प्रकार 5 की कुल संख्या = 12 + 1 (25 से अतिरिक्त) + 1 (50 से अतिरिक्त) = 14। अतः अंत में 14 शून्य होंगे।
In English (Question & Model Answer)
How many zeroes are there at the end of the following product?
1 × 5 × 10 × 15 × 20 × 25 × 30 × 35 × 40 × 45 × 50 × 55 × 60
Correct Option: (d) 15
To find the number of zeroes at the end of the product, we need to count the total number of factors of 5 in the prime factorization of the given expression, since factors of 2 are abundantly available. The expression is: 1 × 5 × (2 × 5) × (3 × 5) × (4 × 5) × (5 × 5) × (6 × 5) × (7 × 5) × (8 × 5) × (9 × 5) × (10 × 5) × (11 × 5) × (12 × 5). Let's count the exponents of 5 explicitly: 5 has one 5 (total 1). 10 has one 5 (total 2). 15 has one 5 (total 3). 20 has one 5 (total 4). 25 has two 5s (5 × 5) (total 4 + 2 = 6). 30 has one 5 (total 7). 35 has one 5 (total 8). 40 has one 5 (total 9). 45 has one 5 (total 10). 50 has two 5s (2 × 25) (total 10 + 2 = 12). 55 has one 5 (total 13). 60 has one 5 (total 14). Wait, let's recount carefully: Multiples of 5 up to 60 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 (12 multiples). Among these, 25 contributes an extra factor of 5, and 50 contributes an extra factor of 5. Total factors of 5 = 12 + 1 (from 25) + 1 (from 50) = 14 factors of 5. Wait, let's check the options. Option (c) is 14. Let's verify: 12 multiples + 2 extra = 14 zeroes. Thus, correct option index is 2.