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Senior PS Moderator V
Joined: 26 Feb 2016
Posts: 3300
Location: India
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If p is an integer and 1365^p is a factor of 200  [#permalink]

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15 00:00

Difficulty:   75% (hard)

Question Stats: 53% (01:57) correct 47% (02:16) wrong based on 191 sessions

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If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

Source: Experts Global

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examPAL Representative P
Joined: 07 Dec 2017
Posts: 1155
Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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1
1
4
pushpitkc wrote:
If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

Source: Experts Global

As the question explicitly asks about factors and we see no way to apply divisibility rules, we'll start by prime factorization.
This is a Precise approach.

1365 = 1300 + 65 = 5(260 + 13) = 5*273 = 5*3*91=5*3*7*13
Now we'll look to our answers to help us out.
p=1 definitley works so (E) is eliminated.
if p = 7 were true than 200! would need to have at least 7 of each of 3,5,7,13. Since 7*13 = 91 then 200! has all multiples of 13 smaller or equal to 91 (and certainly also those for 3,5,7.)
So (C) works and (D) is eliminated.
Could p = 15? 15*13 =195 which is still smaller than 200! So the first 15 multiples of 13 are in 200! and (B) works.
(C) is eliminated.
Could p = 16? 16*13 = 195+13=208 so 200! does not have 16 different multiples of 13. However, before eliminating (A) we should look out for the square trap - since 169 = 13*13 than it has two factors of 13 and so 200! has 16 total 13s in its factorization - one for each of the 15 multiples and one more for 13^2.
(B) is eliminated.

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Director  P
Joined: 02 Oct 2017
Posts: 710
Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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1365= 3*5*7*13

We need to find value of p
So p will depend on lowest value one can take among factors of 1365

Among factors,13 must have less occurrences than any other factor in 200!

So occurrence of 13 is 16 in 200!
And hence 16 is limiting factor for 1365^p to be factor of 200!

Give kudos if it helps

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Intern  B
Joined: 01 Aug 2012
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Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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1
200! has 200/5 + 200/25 + 200/125 zeroes ~= 49
1365 -> since the answer options are not close -> Approximating as (1000)^[p] -> (10)^[3p]
So 3p<49
p~<16
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Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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pushpitkc wrote:
If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

1365 = 5 x 273 = 5 x 3 x 91 = 5 x 3 x 7 x 13

Since 13 is the largest prime factor of 1365, we see that the maximum value of p will be based on the number of 13’s that are factors of 200!.

We see that 13 is a factor of 200!, and, in fact, every one of the multiples of 13 less than 200 is also a factor of 200!. Since 13 x 15 = 195, we see that there are 15 multiples of 13 that are factors of 200!. A But since 169 = 13^2, we see that there is an additional factor of 13, so there are 16 factors of 13 in 200! Thus, the maximum value of p is 16.

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# Scott Woodbury-Stewart

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Intern  B
Joined: 04 Apr 2017
Posts: 23
Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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ScottTargetTestPrep wrote:
pushpitkc wrote:
If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

1365 = 5 x 273 = 5 x 3 x 91 = 5 x 3 x 7 x 13

Since 13 is the largest prime factor of 1365, we see that the maximum value of p will be based on the number of 13’s that are factors of 200!.

We see that 13 is a factor of 200!, and, in fact, every one of the multiples of 13 less than 200 is also a factor of 200!. Since 13 x 15 = 195, we see that there are 15 multiples of 13 that are factors of 200!.A But since 169 = 13^2, we see that there is an additional factor of 13, so there are 16 factors of 13 in 200! Thus, the maximum value of p is 16.

Hi ScottTargetTestPrep

Sorry, I didn't get the highlighted part.

I'm following the below rule strictly, do you suggest a modification for it?
Attachments rule.JPG [ 17.42 KiB | Viewed 1195 times ]

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If p is an integer and 1365^p is a factor of 200  [#permalink]

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1
hisho wrote:
Hi ScottTargetTestPrep

Sorry, I didn't get the highlighted part.

I'm following the below rule strictly, do you suggest a modification for it?

Hey hisho ,

No, you are not missing anything. The formula you mentioned is the another way to solve this question.

What ScottTargetTestPrep meant was since we will have every factor of 13 less than 200 being a factor of 200! because 200! when expanded will include all those factors. Based on that we calculated there are 15 factors of 13 that are less than 200. All of those when divide 200! will give us an integer.

Now, we do have 169 also as one of it's factor less than 200. But this 169 is $$13^2$$, that means we have two 13's in 169. So, we will add 1 to 15 and hence get the total number of 13s = 16.

$$200/13$$ + $$200/169$$

15 + 1 = 16.

Does that make sense?
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Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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pushpitkc wrote:
If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

Source: Experts Global

The Factors of $$1365$$ are - $$3,5,7,13$$. As 13 is the highest factor for 1365, we need to find numbers from 1 to 200 which are divisible 13.

So there are 16 numbers from 1 to 200 which are divisible by 13 (i.e. 13, 26,......, 169(two counts),... 195).

Hence, 16.
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Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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hisho wrote:
ScottTargetTestPrep wrote:
pushpitkc wrote:
If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

1365 = 5 x 273 = 5 x 3 x 91 = 5 x 3 x 7 x 13

Since 13 is the largest prime factor of 1365, we see that the maximum value of p will be based on the number of 13’s that are factors of 200!.

We see that 13 is a factor of 200!, and, in fact, every one of the multiples of 13 less than 200 is also a factor of 200!. Since 13 x 15 = 195, we see that there are 15 multiples of 13 that are factors of 200!.A But since 169 = 13^2, we see that there is an additional factor of 13, so there are 16 factors of 13 in 200! Thus, the maximum value of p is 16.

Hi ScottTargetTestPrep

Sorry, I didn't get the highlighted part.

I'm following the below rule strictly, do you suggest a modification for it?

When we said there are 15 multiples of 13 less than 200, we included every multiple of 13 from 13 to 195. Thus 169 is also included. However, the difference between 169 and all the other multiples is that 169 = 13^2, where all the other multiples of 13 are 13^1 * k.

Thus, the reason why we said that 169 has an additional factor of 13 is because 169 as two prime factors of 13 as opposed to just 1.
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# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

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Posts: 991
WE: Supply Chain Management (Energy and Utilities)
Re: If p is an integer and 1365^p is a factor of 200  [#permalink]

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pushpitkc wrote:
If p is an integer and $$1365^p$$ is a factor of $$200!$$, what is the maximum possible value of p?

A. 16
B. 15
C. 7
D. 1
E. 0

Source: Experts Global

Let's write 1365 in it's prime factorization form, 1365=3*5*7*13

'p' will be maximum when the power of 13 in 200! is the highest.

Now let's apply the formula for finding number powers of the prime number 13 in 200!, we have

$$\frac{200}{13}$$+$$\frac{200}{13^2}$$
=15+1
=16

hence option(A) is the correct answer.
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Rise above the storm, you will find the sunshine Re: If p is an integer and 1365^p is a factor of 200   [#permalink] 03 Jun 2018, 21:02
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