3. Find the prime factorization of 10!4. Which positive integers less than 12 are relatively prime to 12?5. Which positive integers less than 30 are relatively prime to 30?
Question
- Find the prime factorization of 10!4. Which positive integers less than 12 are relatively prime to 12?5. Which positive integers less than 30 are relatively prime to 30?
Solution
- The prime factorization of 10! (10 factorial) is found by breaking down each number from 10 to 1 into its prime factors.
10 = 2 * 5 9 = 3 * 3 8 = 2 * 2 * 2 7 = 7 6 = 2 * 3 5 = 5 4 = 2 * 2 3 = 3 2 = 2 1 = 1
So, the prime factorization of 10! is 2^8 * 3^4 * 5^2 * 7.
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Two numbers are relatively prime if their greatest common divisor (GCD) is 1. The positive integers less than 12 that are relatively prime to 12 are: 1, 5, 7, 11.
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Similarly, the positive integers less than 30 that are relatively prime to 30 are: 1, 7, 11, 13, 17, 19, 23, 29.
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