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What is the greatest value of x such that 8^x is a factor of 18! ?

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What is the greatest value of x such that 8^x is a factor of 18! ?  [#permalink]

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New post 14 Dec 2016, 04:58
1
5
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

65% (01:21) correct 35% (01:14) wrong based on 105 sessions

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Re: What is the greatest value of x such that 8^x is a factor of 18! ?  [#permalink]

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New post 14 Dec 2016, 08:01
1
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 18! ?

A. 1
B. 2
C. 4
D. 5
E. 6


\(8 = 2^3\)

Highest power of 2 in 18! is 16

18/2 = 9
9/2 = 4
4/2 = 2
2/1 = 1

SO, the highest power of 8 in 18! will be 16/3 = 5

Answer will be (D) 5

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Re: What is the greatest value of x such that 8^x is a factor of 18! ?  [#permalink]

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New post 15 Dec 2016, 17:14
1
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 18! ?

A. 1
B. 2
C. 4
D. 5
E. 6


Since 8 = 2^3, we are actually trying to determine the greatest value of x such that 2^(3x) is a factor of 18!.

Let’s first determine the number of factors of 2 within 18!. To do that, we can use the following shortcut in which we divide 18 by 2, and then divide the quotient of 18/2 by 2 and continue this process until we can no longer get a nonzero integer as the quotient.

18/2 = 9

9/2 = 4 (we can ignore the remainder)

4/2 = 2

2/2 = 1

Since 1/2 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 2 within 18!.

Thus, there are 9 + 4 + 2 + 1 = 16 factors of 2 within 18!

However, we are not asked for the number of factors of 2; instead we are asked for the number of factors of 8. We see that 16 factors of 2 will produce 5 factors of 8.

Note: To clarify the final answer, note that the 16 factors of 2 can be expressed as 2^16. We now must break this number 2^16 into as many factors of 8 as possible; thus, we will have

2^16 = 2^3 x 2^3 x 2^3 x 2^3 x 2^3 x 2^1

2^16 = 8 x 8 x 8 x 8 x 8 x 2

2^16 = 8^5 x 2

Note that we can get only 5 factors of 8 out of 2^16; there is a “leftover” 2 that cannot be used.

Answer: D
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Re: What is the greatest value of x such that 8^x is a factor of 18! ?  [#permalink]

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New post 04 Feb 2019, 01:34
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 18! ?

A. 1
B. 2
C. 4
D. 5
E. 6



8^x = 2^3x

and 18!/2 = 18!/2 + 18!/4+18!/8+18!/16
=> 9+4+2+1 = 16
or say
3x=16
x= 5
IMO D
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What is the greatest value of x such that 8^x is a factor of 18! ?  [#permalink]

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New post 10 Feb 2019, 09:49
1
1. List all even factors of 18: 2, 4, 6, 8, 10, 12, 14, 16, 18.
2. Find powers of 2 in each of these factors: 1, 2, 1, 3, 1, 2, 1, 4, 1.
3. Note that sum all powers of 2 equals 16.
4. Note that \(8^x=2^{3x}\), and it will be a factor of 18! as long as the number of 2s are less or equal to 16, i.e. \(3x\leq{16}\). This gives us \(x\leq{16/3}\) or \(x\leq{5.(3)}\).
5. Scan answers and note that all answers are integers, so the maximum value of x is 5. Answer D.
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Re: What is the greatest value of x such that 8^x is a factor of 18! ?  [#permalink]

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New post 10 Feb 2019, 10:02
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 18! ?

A. 1
B. 2
C. 4
D. 5
E. 6


Trailing zeros logic

18/2 =9
18/4 = 4
18/8 = 2
18/16= 1

16/3 = 5.3 ~ 5

D
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Re: What is the greatest value of x such that 8^x is a factor of 18! ?   [#permalink] 10 Feb 2019, 10:02
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