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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!

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.

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.

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

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18 May 2015, 03:39

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!

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, 00:55

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.

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

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02 Jun 2017, 15:15

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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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, 08:28

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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