## Que: How many different prime factors of $$4^5+4^6+4^7$$ are there?

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### Que: How many different prime factors of $$4^5+4^6+4^7$$ are there?

by [email protected] Revolution » Mon May 10, 2021 9:26 pm

00:00

A

B

C

D

E

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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 [email protected] 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.