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For every positive even integer n, the function h(n) is defined to be
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20 Sep 2021, 09:02
20 Sep 2021, 09:02
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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. Also, \(P\) is the smallest prime factor of \(h(100) +1\).
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
Quantity A |
Quantity B |
\(P\) |
\(50\) |
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
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Posts: 6218
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For every positive even integer n, the function h(n) is defined to be
[#permalink]
20 Sep 2021, 12:17
20 Sep 2021, 12:17
KarunMendiratta wrote:
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. Also, \(P\) is the smallest prime factor of \(h(100) +1\).
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
Quantity A |
Quantity B |
\(P\) |
\(50\) |
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
Important Concept: If integer k is greater than 1, and k is a factor (divisor) of N, then k is not a divisor of N+1
For example, since 7 is a factor of 350, we know that 7 is not a factor of (350+1)
Similarly, since 8 is a factor of 312, we know that 8 is not a factor of 313
Now let’s examine h(100)
h(100) = (2)(4)(6)(8)….(96)(98)(100)
= (2x1)(2x2)(2x3)(2x4)....(2x48)(2x49)(2x50)
Factor out all of the 2's to get: h(100) = [2^50][(1)(2)(3)(4)….(48)(49)(50)]
Since 2 is in the product of h(100), we know that 2 is a factor of h(100), which means that 2 is not a factor of h(100)+1 (based on the above rule)
Similarly, since 3 is in the product of h(100), we know that 3 is a factor of h(100), which means that 3 is not a factor of h(100)+1 (based on the above rule)
Similarly, since 4 is in the product of h(100), we know that 4 is a factor of h(100), which means that 4 is not a factor of h(100)+1 (based on the above rule)
.
.
.
.
Similarly, since 50 is in the product of h(100), we know that 50 is a factor of h(100), which means that 50 is not a factor of h(100)+1 (based on the above rule)
So, we can see that none of the numbers from 2 to 50 can be factors of h(100)+1, which means the smallest prime factor of h(100)+1 must be greater than 50.
In other words, P is greater than 50.
Answer: A
Cheers,
Brent
gmatclubot
For every positive even integer n, the function h(n) is defined to be [#permalink]
20 Sep 2021, 12:17
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