**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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00:00

**A**

**B**

**C**

**D**

**E**

A. 2

B. 3

C. 1

D. 5

E. 6

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=> 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

Answer B

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