Que: How many different prime factors of \(3^5+3^6+3^7\) are there?
A. 2
B. 3
C. 4
D. 5
E. 6
Que: How many different prime factors of \(3^5+3^6+3^7\) are there?
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- Max@Math Revolution
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Solution: Factors of M are the integers dividing M without a remainder => If M=ab (a and b are positive integers) => a and b are factors of M => When M is expressed as a product of its prime factors only
=> We have prime factorized M =>If we prime factorize a positive integer M => M= \(\left(p_1\right)^{t_1}\ \ \cdot\left(p_2\right)^{t_2}\cdot........\cdot\left(p_n\right)^{t_n}\ \ \ \ \)
=> \(p_i\) : Different prime factors and \(t_i\) : Positive integers and the exponents of different prime factors, where i = 1,2,….,n
=> number of prime factors = n
We have to find the number of different factors of \(3^5+3^6+3^7\)
=> \(3^5+3^6+3^7=3^5+3^5\cdot3^1+3^7\cdot3^2\)
=> \(3^5\left(1+3^1+3^2\right)\)
=> \(3^5\left(1+3+9\right)\)
=> \(3^5\cdot13\)
=> ∴ Prime factors 3 and 13
=> ∴ Number of prime factors =2
Therefore, A is the correct answer.
Answer A
=> We have prime factorized M =>If we prime factorize a positive integer M => M= \(\left(p_1\right)^{t_1}\ \ \cdot\left(p_2\right)^{t_2}\cdot........\cdot\left(p_n\right)^{t_n}\ \ \ \ \)
=> \(p_i\) : Different prime factors and \(t_i\) : Positive integers and the exponents of different prime factors, where i = 1,2,….,n
=> number of prime factors = n
We have to find the number of different factors of \(3^5+3^6+3^7\)
=> \(3^5+3^6+3^7=3^5+3^5\cdot3^1+3^7\cdot3^2\)
=> \(3^5\left(1+3^1+3^2\right)\)
=> \(3^5\left(1+3+9\right)\)
=> \(3^5\cdot13\)
=> ∴ Prime factors 3 and 13
=> ∴ Number of prime factors =2
Therefore, A is the correct answer.
Answer A
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