For every positive even integer n, the function h(n)

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) +2, then p is?

A. between 2 and 20
B. between 10 and 20
C. between 20 and 30
D. between 30 and 40
E. 2

Bunuel · Accepted Answer

h(100)+2=2*4*6*...*100+2 . Notice that we can factor out 2 from h(100)+2, thus the smallest prime factor of h(100)+2 is 2: h(100)+2=2*(4*6*...*100+1) . Answer: E. Similar question to practice: for-every-positive-even-integer-n-the-function-h-n-is-126691.html Hope it helps.

score780 · Answer

h(100) will have 0 in its unit digit because while doing the calculation of the function we are multiplying by 10 (and 100). So h(100)+2 will have 2 in the units digit so smallest prime factor is 2.

Shiv2016 · Answer

The answer will be even and thus the SMALLEST prime factor would be 2 (even).

coylahood · Answer

h(100) = (100)(98)(96)...(6)(4)(2)=(2)(50)(2)(49)(2)(48)...(2)(3)(2)(2)(2)(1)=2^50*50!
h(100)+2= 2^50*50!+2
2 is the smallest prime number for which h(100)+2 results in a remainder of 0. Therefore E is the correct answer.

gmatway · Answer

Hi, Can you elaborate this a little bit . I am confused between how do we decide smallest /largest prime no. in such cases . I HAD ATTEMPT A PREVIOUS question by your method and I thought 2 is smallest even prime no. but that case turned out to be different. below is the link of that question https://gmatclub.com/forum/for-every-po ... 12521.html

coylahood · Answer

h(100)+2 is an even number and every even number must be divisible by 2. According to number theory, 2 is the only even prime number and it is also the smallest prime number. Since h(100)+2 is divisible by 2, then 2 must be the smallest prime factor of h(100)+2.

asmba · Answer

The Q says that 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) +2, then p is:
p has to be 2 as 2 is the only even prime number and it is part of the set.

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For every positive even integer n, the function h(n)

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For every positive even integer n, the function h(n)

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