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If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors?
- a)2
- b)3
- c)5
- d)6
Correct answer is option 'B'. Can you explain this answer?
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If n is a positive integer less than 200 and 14n/60 is an integer, the...
To find the number of different positive prime factors of n, we need to determine the prime factorization of 14n/60.
Prime factorization of 14n/60:
- First, let's simplify the fraction by canceling out common factors. Both 14 and 60 are divisible by 2, so we can divide both by 2:
14n/60 = 7n/30
Now, to determine the prime factorization of 7n/30, we need to find the prime factors of both 7 and 30.
Prime factorization of 7:
7 is a prime number, so its prime factorization is simply 7.
Prime factorization of 30:
- First, let's find the prime factorization of 30 by dividing it by its smallest prime factor, which is 2:
30 ÷ 2 = 15
- Now, let's continue dividing the result by its smallest prime factor, which is 3:
15 ÷ 3 = 5
So, the prime factorization of 30 is 2 × 3 × 5.
Prime factorization of 7n/30:
Now, let's combine the prime factorizations of 7 and 30 to find the prime factorization of 7n/30:
7n/30 = 7 × (n/2) × (1/3) × 5
From this prime factorization, we can see that the factors of n are independent of the factors of 7 × (1/2) × (1/3) × 5. This means that the prime factors of n could be any prime numbers other than 2, 3, and 5. Therefore, n can have at most 3 different positive prime factors.
Hence, the correct answer is option B) 3.
Prime factorization of 14n/60:
- First, let's simplify the fraction by canceling out common factors. Both 14 and 60 are divisible by 2, so we can divide both by 2:
14n/60 = 7n/30
Now, to determine the prime factorization of 7n/30, we need to find the prime factors of both 7 and 30.
Prime factorization of 7:
7 is a prime number, so its prime factorization is simply 7.
Prime factorization of 30:
- First, let's find the prime factorization of 30 by dividing it by its smallest prime factor, which is 2:
30 ÷ 2 = 15
- Now, let's continue dividing the result by its smallest prime factor, which is 3:
15 ÷ 3 = 5
So, the prime factorization of 30 is 2 × 3 × 5.
Prime factorization of 7n/30:
Now, let's combine the prime factorizations of 7 and 30 to find the prime factorization of 7n/30:
7n/30 = 7 × (n/2) × (1/3) × 5
From this prime factorization, we can see that the factors of n are independent of the factors of 7 × (1/2) × (1/3) × 5. This means that the prime factors of n could be any prime numbers other than 2, 3, and 5. Therefore, n can have at most 3 different positive prime factors.
Hence, the correct answer is option B) 3.
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