標題:
GMAT Quantitative Question
發問:
For every positive interger n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is a) between 2 and 10 b) between 10 and 20 c) between 20 and 30 d) between 30 and 40 e) greater than 40
最佳解答:
The answer is e. h(100) = 2 4 6 8 ........ 98 100 h(100) = 2 x (1 2 3 4 ........ 49 50) h(100) is a multiple of 2, 3, 4, 5 ........ , 49, 50. h(100) + 1 is not a multiple of 2, 3, 4, 5 ........ , 49, 50. Hence, the smallest factor (no matter whether it is prime or not) of h(100) + 1 must be greater than 50. Thus, e is the best answer. =
其他解答:
h(n)=2x4x6x..xn (total n/2 terms) h(n)=(2^n/2)(1x2x3..x(n/2)) h(100)+1 = (2^50)(1x2x3x..x50)+1 so if there is a prime factor for this expression, this prime factor has to be greated than 50. It is because if it is less than 50, it is a factor of the first term on the right hand side but not the second term (ie. 1). So the answer is (e)
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