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02 Jan 2015, 13:10
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Hi,
This question tends to trouble most Test Takers and it is definitely tougher than a typical GMAT question.
As a general rule, Quant questions are almost always based on a pattern of some kind (math formula, math rule, Number Property, etc.). If you can't immediately deduce a pattern, then you might have to "play around" a bit with the question to try to deduce what the pattern is. In the broad sense, it's critical thinking: here's a weird situation - what can I do to figure it out?
Based on the description of the function in the prompt, we can run some "TESTS" to try to figure things out….
The H(n) is the product of all the even integers from 2 to n, inclusive.
So….
H(4) = 2x4 = 8
The prime factors of 8 are (2)(2)(2)
If we do H(4) + 1 = 9, then the prime factors are (3)(3)
NOTICE how NONE of the prime factors of 8 are in 9? That's interesting….
H(6) = 2x4x6 = 48
The prime factors of 48 are (2)(2)(2)(2)(3)
If we do H(6) + 1 = 49, then the prime factors are (7)(7)
NOTICE how NONE of the prime factors of 48 are in 49? That's interesting….and probably a pattern, since it's happened TWICE NOW.
From here, I'd have to deduce that this pattern holds true. With H(100), I know that there are LOTS of primes that go in (the largest of which is 47, which can be "found" in 94). I have to assume that NONE of them will go into H(100) + 1. Thus, the smallest prime would have to be greater than 47. The question doesn't actually ask us for the exact prime number though - it just asked for the range that the prime would fall into.
Final Answer:
The takeaway from all of this is that you shouldn't be afraid to "play" with a question a bit. In the end, you don't have to be a brilliant mathematician to answer this question, but you're also not allowed to just stare at it either.
GMAT assassins aren't born, they're made,
Rich
This question tends to trouble most Test Takers and it is definitely tougher than a typical GMAT question.
As a general rule, Quant questions are almost always based on a pattern of some kind (math formula, math rule, Number Property, etc.). If you can't immediately deduce a pattern, then you might have to "play around" a bit with the question to try to deduce what the pattern is. In the broad sense, it's critical thinking: here's a weird situation - what can I do to figure it out?
Based on the description of the function in the prompt, we can run some "TESTS" to try to figure things out….
The H(n) is the product of all the even integers from 2 to n, inclusive.
So….
H(4) = 2x4 = 8
The prime factors of 8 are (2)(2)(2)
If we do H(4) + 1 = 9, then the prime factors are (3)(3)
NOTICE how NONE of the prime factors of 8 are in 9? That's interesting….
H(6) = 2x4x6 = 48
The prime factors of 48 are (2)(2)(2)(2)(3)
If we do H(6) + 1 = 49, then the prime factors are (7)(7)
NOTICE how NONE of the prime factors of 48 are in 49? That's interesting….and probably a pattern, since it's happened TWICE NOW.
From here, I'd have to deduce that this pattern holds true. With H(100), I know that there are LOTS of primes that go in (the largest of which is 47, which can be "found" in 94). I have to assume that NONE of them will go into H(100) + 1. Thus, the smallest prime would have to be greater than 47. The question doesn't actually ask us for the exact prime number though - it just asked for the range that the prime would fall into.
Final Answer:
The takeaway from all of this is that you shouldn't be afraid to "play" with a question a bit. In the end, you don't have to be a brilliant mathematician to answer this question, but you're also not allowed to just stare at it either.
GMAT assassins aren't born, they're made,
Rich
26 Feb 2015, 13:36
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Hi All,
I've been asked to post this solution here, so here's another way to handle this question:
For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of H(100) + 1, then p is:
1. between 2 and 10
2. between 10 and 20
3. between 20 and 30
4. between 30 and 40
5. greater than 40
The main idea behind this prompt is:
"The ONLY number that will divide into X and (X+1) is 1."
In other words, NONE of the factors of X will be factors of X+1, EXCEPT for the number 1.
Here are some examples:
X = 2
X+1 = 3
Factors of 2: 1 and 2
Factors of 3: 1 and 3
ONLY the number 1 is a factor of both.
X = 9
X+1 = 10
Factors of 9: 1, 3 and 9
Factors of 10: 1, 2, 5 and 10
ONLY the number 1 is a factor of both.
Etc.
Since the H(100) is (100)(98)(96)....(4)(2)....we can deduce....
1) This product will have LOTS of different factors
2) NONE of those factors will divide into H(100) + 1.
H(100) contains all of the primes from 2 through 47, inclusive (the 47 can be "found" in the "94"), so NONE of those will be in H(100) + 1. We don't even have to calculate which prime factor is smallest in H(100) + 1; we know that it MUST be a prime greater than 47....and there's only one answer that fits.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
I've been asked to post this solution here, so here's another way to handle this question:
For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of H(100) + 1, then p is:
1. between 2 and 10
2. between 10 and 20
3. between 20 and 30
4. between 30 and 40
5. greater than 40
The main idea behind this prompt is:
"The ONLY number that will divide into X and (X+1) is 1."
In other words, NONE of the factors of X will be factors of X+1, EXCEPT for the number 1.
Here are some examples:
X = 2
X+1 = 3
Factors of 2: 1 and 2
Factors of 3: 1 and 3
ONLY the number 1 is a factor of both.
X = 9
X+1 = 10
Factors of 9: 1, 3 and 9
Factors of 10: 1, 2, 5 and 10
ONLY the number 1 is a factor of both.
Etc.
Since the H(100) is (100)(98)(96)....(4)(2)....we can deduce....
1) This product will have LOTS of different factors
2) NONE of those factors will divide into H(100) + 1.
H(100) contains all of the primes from 2 through 47, inclusive (the 47 can be "found" in the "94"), so NONE of those will be in H(100) + 1. We don't even have to calculate which prime factor is smallest in H(100) + 1; we know that it MUST be a prime greater than 47....and there's only one answer that fits.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
Updated on: 24 Jan 2020, 23:13
Originally posted by GMATinsight on 12 Nov 2015, 09:06.
Last edited by GMATinsight on 24 Jan 2020, 23:13, edited 1 time in total.
Last edited by GMATinsight on 24 Jan 2020, 23:13, edited 1 time in total.
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enigma123
As per the definition of the question
h(100) = 2 x 4 x 6 x 8 x 10 x 12 x 14 ... and so on...98 x 100 (Total 50 terms)
=> h(100) = 2^50 (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10......and so on...x 48 x 49 x 50)
This means h(100) is multiple of all prime numbers between 1 and 50
therefore h(100)+1 will leave a remainder of 1 when divided by any prime number from 1 to 50
therefore, p, which is a factor of h(100)+1, will certainly be greater than a prime numbers greater than 50
Hence, "p" must be greater than 40 as per the following options
Answer: Option E
