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Que: How many different prime factors of \(4^5+4^6+4^7\) are there?

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by Max@Math Revolution » Mon May 10, 2021 9:26 pm

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Que: How many different prime factors of \(4^5+4^6+4^7\) are there?

A. 2
B. 3
C. 1
D. 5
E. 6
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Re: Que: How many different prime factors of \(4^5+4^6+4^7\) are there?

by Max@Math Revolution » Thu May 13, 2021 9:24 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 = \(\left(p_1\right)^{t_1}\ \ \ \cdot\ \ \left(p_2\right)^{t_2}\ \ \ ......\ \left(p_n\right)^{t_n}\)

=> \(p_i\): 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 \(4^5+4^6+4^7\)

=> \(4^5+4^6+4^7\) = \(4^5+4^5\cdot4^1+4^5\cdot4^2\)

=> \(4^5\left(1+4^1+4^2\right)\)

=> \(4^5\left(1+4+16\right)\)

=> \(4^5\cdot21\)

=> \(\left(2^2\right)^5\cdot3\cdot7\)

=> ∴ Prime factors 2, 3 and 7

=> ∴ Number of prime factors =3

Therefore, B is the correct answer.

Answer B
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