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# X is the largest prime number less than positive integer N. P is an in

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e-GMAT Representative
Joined: 04 Jan 2015
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X is the largest prime number less than positive integer N. P is an in  [#permalink]

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Updated on: 07 Aug 2018, 02:09
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65% (hard)

Question Stats:

67% (02:16) correct 33% (05:36) wrong based on 455 sessions

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Question 1 of the e-GMAT Primes Trio: 3 Questions on Number of factors and prime factors

X is the largest prime number less than positive integer N. P is an integer such that P = X – 16. Also, Z = 1*2*…*$$\sqrt{P}$$. If N is the first non-zero perfect square whose tens digit and units digit are same, How many different prime factors does Z have?

A. 4
B. 5
C. 6
D. 160
E. 320

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Originally posted by EgmatQuantExpert on 04 May 2015, 05:43.
Last edited by EgmatQuantExpert on 07 Aug 2018, 02:09, edited 6 times in total.
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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04 May 2015, 06:25
5
If N is the first perfect square whose tens digit and units digit are same, then N=100.

X is the largest prime number less than positive integer N , then X is
97.

P is an integer such that P = X – 16.

So P=81

Z=(P^1/2)! = 9!

How many different prime factors does Z have?

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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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05 May 2015, 23:18
1
1
Here is the solution for this question.

Given: We are given that $$X$$ is the largest prime number less than positive integer $$N$$ and that $$N$$ is the first perfect square whose tens digit and units digit are same.

We are also given that $$Z = 1 * 2 * 3 * … * \sqrt{P}$$, where $$P = X – 16$$

The question asks us to find the number of prime factors of $$Z$$.

Approach: Since $$Z$$ is of the form of a factorial, q! in this case. (where $$q = \sqrt{P}$$), the number of prime factors of $$Z$$ will be simply the number of prime numbers less than or equal to $$\sqrt{P}$$.

(For example, $$4!$$ is simply $$1*2*3*4 = 1*2*3*2^2$$. As you can see the only prime factors of $$4!$$ are $$2$$ and $$3$$ which are essentially the prime numbers less than or equal to $$4$$ itself.)

Once we find the value of $$\sqrt{P}$$, the problem simply boils down to counting the number of prime numbers less than $$\sqrt{P}$$.

To find $$\sqrt{P}$$, we need the value of $$P$$ and to find the value of $$P$$, we need the value of $$X$$. (Since $$P = X – 16$$)

Since $$X$$ is the largest prime number less than $$N$$, to find the value of $$X$$, we simply need the value of $$N$$.

$$N$$ has two constraints over it.

a. $$N$$ is a non-zero perfect square.
b. The tens digit of $$N$$ = Units digit of $$N$$

Using these constraints, let us try to determine the value of $$N$$ and then work backwards to solve the problem.

Working Out: Since $$N$$ is a non-zero perfect square, possible values of $$N$$ are: $$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121$$

Notice that $$100$$ is the first value that satisfies the condition that the tens digit and units digit of $$N$$ must be same.

Therefore $$N = 100$$.

Since $$X$$ is the largest prime number less than $$100$$, $$X$$ must be $$97$$.

This gives us $$P = 97 – 16 = 81$$ which in turn gives us $$\sqrt{P} = 9$$.

This means $$Z$$ is simply $$1*2*…*9 = 9!$$

Therefore prime factors of $$Z$$ are simply the prime numbers less than or equal to $$9$$, which are $$2, 3, 5, and 7$$.

Therefore $$Z$$ has $$4$$ different prime factors.

Foot Note: If you are unclear about how the number of different prime factors of n! is same as number of prime numbers less than or equal to n itself, then you can refer to this post.
if-n-is-the-product-of-the-integers-from-1-to-8-inclusive-135542.html#p1522399

Here is another question that tests your conceptual understanding of primes and factors.
x-y-are-integers-find-the-number-of-even-factors-of-4x-197375.html

Regards,
Krishna
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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05 May 2015, 23:27
1
X is the largest prime number less than positive integer N. If N is the first non-zero perfect square whose tens digit and units digit are same.

N ~ (25,36...81,100) = 100 Unit/Ten digit same

N= 100 , X largest prime less than 100 = 97 ; X = 97

P is an integer such that P = X – 16.
P = 97 - 16 = 81

Also, Z = 1*2*…*$$\sqrt{P}$$. , How many different prime factors does Z have?

Z = 1*2*…*$$\sqrt{81}$$.

Z = 1*2*3*4*5*6*7*8*9 = prime { 2,3,5,7} ~ 4

Ans : A
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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07 May 2015, 10:52
One Doubt ... The question says that N is the first non zero perfect square ... 100 has zero in them .. shoudnt N be 144???
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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17 May 2015, 22:58
Udai wrote:
One Doubt ... The question says that N is the first non zero perfect square... 100 has zero in them .. shoudnt N be 144???

Dear Udai

By non-zero perfect square, we mean a perfect square whose magnitude is not equal to zero. Yes, 100 does have 2 zeroes in it, but is the magnitude of 100 equal to zero? Not at all! Therefore, 100 doesn't violate the 'non-zero perfect square' bit.

Hope this helped!

Japinder
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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18 May 2015, 04:39
EgmatQuantExpert wrote:
X is the largest prime number less than positive integer N. P is an integer such that P = X – 16. Also, Z = 1*2*…*$$\sqrt{P}$$. If N is the first non-zero perfect square whose tens digit and units digit are same, How many different prime factors does Z have?

A. 4
B. 5
C. 6
D. 160
E. 320

Here is a fresh question from the e-GMAT bakery! Go ahead and give it a shot!

This is Question 1 of the e-GMAT Primes Trio: 3 Questions on Number of factors and prime factors

Regards,
The e-GMAT Quant Team

P.S.: Solutions with clarity of thought and elegance will get kudos!

Would you say that this is a 650 level question? I got confused when reading the prompt on what I was suppose to do, but once I figured out what it was asking the problem only took me 1:45.
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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27 Mar 2016, 01:55
I was able to derive all information from the question prompt except that Z = 1*2*…*√P denotes the factorial of √P. Hence my question is:
Is Z = 1*2*…*√P a common way of denoting the factorial of √P? I have seen the factorial be denoted √P!, but I have never seen this denotation before.
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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22 Aug 2017, 04:11
Pretty straight forward question.
However took around 3 minutes to solve.
But, I love the way e-gmat has built their course around such concepts.
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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03 Sep 2017, 05:27
@e-gmat, abhimahna, and Bunuel

If it was not given that N is a non-zero integer, then could the value of N be 0?
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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03 Sep 2017, 09:12
Shiv2016 wrote:
@e-gmat, abhimahna, and Bunuel

If it was not given that N is a non-zero integer, then could the value of N be 0?

Hi Shiv2016 ,

We are already given that N is a positive integer, which means N > 0

Hence, even if we were not given N is a first non zero perfect square, we cannot take N = 0

Does that make sense?
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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03 Sep 2017, 09:26
abhimahna wrote:
Shiv2016 wrote:
@e-gmat, abhimahna, and Bunuel

If it was not given that N is a non-zero integer, then could the value of N be 0?

Hi Shiv2016 ,

We are already given that N is a positive integer, which means N > 0

Hence, even if we were not given N is a first non zero perfect square, we cannot take N = 0

Does that make sense?

If that information was also not given, then N could be anything - even 0?
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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03 Sep 2017, 09:28
Shiv2016 wrote:
If that information was also not given, then N could be anything - even 0?

Yes, if we are not given anything about N, it could be +ve number, -ve number or zero. But yes, this question cannot be solved in that case.
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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03 Sep 2017, 10:14
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Re: X is the largest prime number less than positive integer N. P is an in  [#permalink]

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