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# What is the greatest value of the positive integer p such that 2^p is

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Math Expert
Joined: 02 Sep 2009
Posts: 44566
What is the greatest value of the positive integer p such that 2^p is [#permalink]

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01 Apr 2018, 10:36
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What is the greatest value of the positive integer p such that 2^p is a factor of 80^50?

(A) 4
(B) 10
(C) 50
(D) 100
(E) 200
[Reveal] Spoiler: OA

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Joined: 05 Feb 2016
Posts: 107
Location: India
Concentration: General Management, Marketing
WE: Information Technology (Computer Software)
Re: What is the greatest value of the positive integer p such that 2^p is [#permalink]

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01 Apr 2018, 10:43
Bunuel wrote:
What is the greatest value of the positive integer p such that 2^p is a factor of 80^50?

(A) 4
(B) 10
(C) 50
(D) 100
(E) 200

80^50=(2^200*5^50)

so greatest value is 200

option E
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Concentration: Accounting, Finance
GPA: 3.68
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What is the greatest value of the positive integer p such that 2^p is [#permalink]

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Updated on: 07 Apr 2018, 15:20
Bunuel wrote:
What is the greatest value of the positive integer p such that 2^p is a factor of 80^50?

(A) 4
(B) 10
(C) 50
(D) 100
(E) 200

we are asked to find the greatest value of the positive integer p. In order to find out the greatest value of p as a exponent of base 2 we have to factorize 80^50.

(2*2*2*2*5)^50
={(2^4)*5}^50
=(2^200 )*5^50

so, 2^200 shows that the greatest value of p. The correct answer is E.

Originally posted by selim on 06 Apr 2018, 13:39.
Last edited by selim on 07 Apr 2018, 15:20, edited 1 time in total.
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What is the greatest value of the positive integer p such that 2^p is [#permalink]

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07 Apr 2018, 12:15
Bunuel wrote:
What is the greatest value of the positive integer p such that 2^p is a factor of 80^50?

(A) 4
(B) 10
(C) 50
(D) 100
(E) 200

$$(ab)^{n} = a^{n}b^{n}$$
$$(a^{m})^{n}=a^{(m*n)}$$

Find prime factors of 80, raise to 50th power
$$80 = (2*2*2*2*5) = (2^4*5)$$
$$(2^4*5)^{50}=$$
$$(2^4)^{50}*5^{50}=$$
$$2^{200}5^{50}$$

5 to any power ends in 5. NEVER divisible by 2. The other factor of $$(80)^{50}=2^{200}$$

$$(80)^{50}=$$
$$(2^{200}*5^{50})=(2^{p}*5^{50})$$
$$2^{p}=2^{200}$$
$$p=200$$

There are 200 powers of 2 in $$(80)^{50}$$

selim , I think there is a small typo in your answer: $$5^{200}$$
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What is the greatest value of the positive integer p such that 2^p is   [#permalink] 07 Apr 2018, 12:15
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