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Food for thought..

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Food for thought..

by vittalgmat » Fri Jan 09, 2009 11:33 pm
Hi,

While trying to solve this problem in the link below, I thought of this question.
wonder if this exists.

Can u have a number X such that it has n factors and all the factors are prime numbers. Assume n > 2.

Is there such a number?

This question arose, coz I was trying to play with the question setup in stmt 2
of the link below.

https://www.beatthegmat.com/gmat-prep-di ... 17112.html

Thanks
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by cramya » Sat Jan 10, 2009 12:03 am
Good question & hope I understood it correctly.

I dont think that can be the case.

IMO this is the reason :

If there were more than 1 prime as a factor for the number X then the product of 2 primes will be a non prime. A number X would always have 1 as a factor so the number X with just primes as factors cannot exist.

Regards,
Cramya
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Re: Food for thought..

by Ian Stewart » Sat Jan 10, 2009 12:05 am
vittalgmat wrote:
Can u have a number X such that it has n factors and all the factors are prime numbers. Assume n > 2.

Is there such a number?
There is no positive integer in the world which only has prime factors and no other factors - every number has 1 as a factor, and 1 is not prime. And, any number that has a few prime divisors will have a lot of non-prime factors. For example, say your number is equal to the product of three primes, p, q and r:

n = pqr

If you prefer to think of real numbers, n might be 30, or 105. Well, n certainly has three prime factors, p, q and r, but it has five other factors: 1, pq, pr, qr and pqr (or in the case of 30, we have the factors 2, 3 and 5, but also 1, 6, 10, 15 and 30).

Because of this, no number has more prime divisors than non-prime divisors. Some numbers have an equal number of prime and non-prime divisors; any prime p, for example, has one prime divisor (p) and one non-prime divisor (1). This suggests a possibly interesting question: do any other numbers (besides prime numbers) have an equal number of prime and non-prime divisors? I'll leave that question open for the moment.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by cramya » Sat Jan 10, 2009 12:06 am
The easiest way to set up statement II would be

k = 2*3*5*7*2*3*5*7

YES

k = 2*3*5*7*2

NO

Combined

k = 2*3*5*7*2 NO
k= 2*3*5*7*2*3*5*7 YES

Hence E)

Hope this helps!

Regards,
CR
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by cramya » Sat Jan 10, 2009 12:13 am
do any other numbers (besides prime numbers) have an equal number of prime and non-prime divisors? I'll leave that question open for the moment
Ian,
Can we take it that u r asking distinct prime factors and distinct non prime factors? The count of each are equal.

Just wanted to make sure I understood the question correctly.

Regards,
Cramya
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by cramya » Sat Jan 10, 2009 12:19 am
On the assumption that u meant distinct:

6 has 1 and 6 as non prime factors
2,3 as its prime factors

Ian,

The numbers that qualify for your question would it be product of just 2 primes, Am I anywhere close to the answer?

Regards,
CR
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by vittalgmat » Sat Jan 10, 2009 12:34 am
Thanks Ian and Cramya,
I tried a few numbers and kinda arrived at what Ian mentioned, although not so succinctly. But I was not sure of the generalization.

greatly appreciate it.

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by cramya » Sat Jan 10, 2009 12:43 am
No probs Vittal! Anytime. More than the solution it's the various questions posed that sometimes help us learn more...

Regards,
CR
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by Ian Stewart » Sat Jan 10, 2009 1:03 am
cramya wrote:On the assumption that u meant distinct:

6 has 1 and 6 as non prime factors
2,3 as its prime factors

Ian,

The numbers that qualify for your question would it be product of just 2 primes, Am I anywhere close to the answer?

Regards,
CR
You're not only close; that's the answer. A product of two different primes pq will always have two prime divisors, p and q, and two non-prime divisors, 1 and pq. If, however, you take the product of three primes, as we saw above you'll have 2^3 = 8 divisors, and only 3 prime divisors. Further, if you take the product of n primes, you'll have 2^n divisors (this should be clear if you know how to count how many divisors a number has from a prime factorization), and only n primes, and n is less than half of 2^n if n is greater than 2. So if you only consider numbers which are equal to the product of three or more distinct primes, you'll have more non-prime divisors than prime divisors.

We haven't considered raising primes to powers, but if you do that, hopefully it's clear that you'll get even more non-prime divisors (for example, p^3 will have only one prime divisor, but has three non-prime divisors: 1, p^2 and p^3), so the only numbers with an equal number of prime and non-prime divisors are either prime or equal to the product of two primes.

Perhaps a bit abstract, but if you understand the above, you're likely in good shape for the higher level number theory that can appear on the GMAT.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by cramya » Sat Jan 10, 2009 1:16 am
Ian,

I bet no GMAT book will have this info, so thanks so much for the valuable information above.

Much appreciated.

Regards,
Cramya
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by vittalgmat » Sat Jan 10, 2009 2:27 am
Ian Stewart wrote:
cramya wrote:On the assumption that u meant distinct:

6 has 1 and 6 as non prime factors
2,3 as its prime factors

Ian,

The numbers that qualify for your question would it be product of just 2 primes, Am I anywhere close to the answer?

Regards,
CR
You're not only close; that's the answer. A product of two different primes pq will always have two prime divisors, p and q, and two non-prime divisors, 1 and pq. If, however, you take the product of three primes, as we saw above you'll have 2^3 = 8 divisors, and only 3 prime divisors. Further, if you take the product of n primes, you'll have 2^n divisors (this should be clear if you know how to count how many divisors a number has from a prime factorization), and only n primes, and n is less than half of 2^n if n is greater than 2. So if you only consider numbers which are equal to the product of three or more distinct primes, you'll have more non-prime divisors than prime divisors.

We haven't considered raising primes to powers, but if you do that, hopefully it's clear that you'll get even more non-prime divisors (for example, p^3 will have only one prime divisor, but has three non-prime divisors: 1, p^2 and p^3), so the only numbers with an equal number of prime and non-prime divisors are either prime or equal to the product of two primes.

Perhaps a bit abstract, but if you understand the above, you're likely in good shape for the higher level number theory that can appear on the GMAT.

Wow!!! that was a goldmine of information, Ian.
Thanks.
While reading, I felt as if I was seeing a DS question for each sentence you wrote; each q in the 700+ range. Implementing that would be sadistic!!!.

thanks a lot
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