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Which of the following is a prime factor of 5^8 - 2^8 ?

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Which of the following is a prime factor of 5^8 - 2^8 ?  [#permalink]

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New post 29 Mar 2018, 00:44
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

64% (01:36) correct 36% (01:36) wrong based on 89 sessions

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Which of the following is a prime factor of 5^8 - 2^8 ?  [#permalink]

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New post 29 Mar 2018, 01:11
Bunuel wrote:
Which of the following is a prime factor of 5^8 - 2^8 ?

I. 2
II. 3
III. 7

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III


Rule: \(a^2 - b^2 = (a+b)(a-b)\)

\(5^8 - 2^8\) can be re-written as \((5^4)^2 - (2^4)^2\)

Further simplifying the expression, we get
\((5^4)^2 - (2^4)^2 = (5^4 + 2^4)(5^4 - 2^4) = (625+16)(5^2 + 2^2)(5^2 - 2^2) = (641)(29)(21)\)

\(5^8 - 2^8\) is divisible by prime number(s) 3 and 7(but not 2) and Option D(II and III) is our answer!
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Re: Which of the following is a prime factor of 5^8 - 2^8 ?  [#permalink]

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New post 29 Mar 2018, 01:12
Bunuel wrote:
Which of the following is a prime factor of 5^8 - 2^8 ?

I. 2
II. 3
III. 7

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III


\(5^8 - 2^8 = (5^4 + 2^4)(5^4 - 2^4)\) .... \(a^2 - b^2 = (a + b)(a - b)\)

\(= (5^4 + 2^4)*(5^2 + 2^2)*(5^2 - 2^2)\)

\(= (5^4 + 2^4)*(5^2 + 2^2) * (5 + 2) ( 5 - 2)\)

hence 7 and 3 are prime factors of the given number.

Also note that for all other factors... we are adding an odd number ( power of 5) to an even number ( power of 2) ... hence it is always odd.

Hence 2 is not a prime factor of given number.

Hence Option (D) is correct.

Best,
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Re: Which of the following is a prime factor of 5^8 - 2^8 ?  [#permalink]

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New post 29 Mar 2018, 06:25
1
Bunuel wrote:
Which of the following is a prime factor of 5^8 - 2^8 ?

I. 2
II. 3
III. 7

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III


5^8 - 2^8 is similar to a^n - b^n
Pls note that (a^n - b^n) is always divisible by (a-b); if n is even (a^n - b^n) is always divisible by (a+b)

Here: a+b= 7 and a-b= 3 are the factors(both prime) of 5^8 - 2^8.
Also note that, 5^8 is an odd no. and 2^8 is an even no. and therefore their subtraction will be an odd no. hence 2 cant be the factor

Ans. D
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Re: Which of the following is a prime factor of 5^8 - 2^8 ?  [#permalink]

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New post 30 Mar 2018, 10:54
Bunuel wrote:
Which of the following is a prime factor of 5^8 - 2^8 ?

I. 2
II. 3
III. 7

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III


Using the difference of squares, we have:

(5^4 - 2^4)(5^4 + 2^4) = (5^2 - 2^2)(5^2 + 2^2)(5^4 + 2^4) = 21 x 29 x 641

We can quickly see that 21 x 29 x 641 does not contain a prime factor of 2.

We can also see that 21 is divisible by both 3 and 7, so 3 and 7 are both prime factors of 5^8 - 2^8. 7.

Answer: D
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Re: Which of the following is a prime factor of 5^8 - 2^8 ?   [#permalink] 30 Mar 2018, 10:54
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