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prime nos..

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by rohitjjw » Sun Sep 05, 2010 12:08 am
for every positive integer n, the function h(n) is defined to be the product of all the even integer from 2 to n, inclusive. if p is the smallest prime factor of h(100)+1 then p is....

1. b/w 2 to 10
2. b/w 10 to 20
3. b/w 20 to 30
4. b/w 30 to 40
5. more than 40

plz tell me hw to solve this quest step by step....
and if i have to find exact no. then how to solve it.........
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Source: — Problem Solving |

by Rahul@gurome » Sun Sep 05, 2010 5:53 am
h(100) = 2 × 4 × 6 ×.... ×100 = (2^50) (1 × 2 × 3 ×.... ×50)
h(100) + 1 = (2^50) (1 × 2 × 3 ×.... ×50) + 1
(2^50) (1 × 2 × 3 ×.... ×50) is divisible by any integer between 1 and 50.
So, (2^50) (1 × 2 × 3 ×.... ×50) + 1 will leave a remainder 1 when divided by any integer between 1 and 50. So, h(100) + 1 cannot have any prime factor less than 50.

The correct answer is [spoiler](E)[/spoiler].
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