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If n is a positive integer greater than 2, what is the greatest prime factor of 3n + 3n + 3n – 3n-2?
- a)3
- b)5
- c)7
- d)11
- e)13
Correct answer is option 'E'. Can you explain this answer?
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If n is a positive integer greater than 2, what is the greatest prime ...
Solution:
Firstly, we can simplify the given expression by combining the exponents:
3n 3n 3n 3n-2 = 33n-2 * 33n = 36n
Now, to find the greatest prime factor of 36n, we can factorize it into prime factors:
36n = 2^2 * 3^2 * n
The greatest prime factor of 36n would be the largest prime factor of n. Since n is greater than 2, we know that it is either a prime number or a composite number with prime factors.
To find the greatest prime factor of n, we can start by dividing n by 2 repeatedly until we get an odd number. For example, if n is 60, we can divide it by 2 three times to get 15:
60 ÷ 2 = 30
30 ÷ 2 = 15
Now, we can check if 15 is a prime number or if it has any other prime factors. We can do this by dividing 15 by the smallest prime numbers, which are 2, 3, 5, 7, 11, 13, etc.
15 ÷ 3 = 5
Since 5 is a prime number, it is the greatest prime factor of n. Therefore, the greatest prime factor of 36n is 13, which is the largest prime factor of 3.
Firstly, we can simplify the given expression by combining the exponents:
3n 3n 3n 3n-2 = 33n-2 * 33n = 36n
Now, to find the greatest prime factor of 36n, we can factorize it into prime factors:
36n = 2^2 * 3^2 * n
The greatest prime factor of 36n would be the largest prime factor of n. Since n is greater than 2, we know that it is either a prime number or a composite number with prime factors.
To find the greatest prime factor of n, we can start by dividing n by 2 repeatedly until we get an odd number. For example, if n is 60, we can divide it by 2 three times to get 15:
60 ÷ 2 = 30
30 ÷ 2 = 15
Now, we can check if 15 is a prime number or if it has any other prime factors. We can do this by dividing 15 by the smallest prime numbers, which are 2, 3, 5, 7, 11, 13, etc.
15 ÷ 3 = 5
Since 5 is a prime number, it is the greatest prime factor of n. Therefore, the greatest prime factor of 36n is 13, which is the largest prime factor of 3.
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