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Que: How many different prime factors of \(3^5+3^6+3^7\) are there?

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by Max@Math Revolution » Thu Mar 18, 2021 9:28 pm

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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
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Re: Que: How many different prime factors of \(3^5+3^6+3^7\) are there?

by Max@Math Revolution » Wed Apr 07, 2021 9:17 pm
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
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