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h is an integer greater than 1,is h a prime number? 1) each

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Director
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h is an integer greater than 1,is h a prime number? 1) each [#permalink] New post 28 Dec 2007, 21:02
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C
D
E

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h is an integer greater than 1,is h a prime number?

1) each prime factor is greater than square root h
2) each prime factor is greater than h/2

Prove OA is D.
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 [#permalink] New post 28 Dec 2007, 23:55
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Let h is not a prime number.

h=p1*p2*....*pn, where p1,...pn- prime numbers

1. h = p1*p2*....*pn > √h*√h*...*√h = h^(n/2) ==>
h > h^(n/2) ==> h^(n/2-1) < 1 ==> n = 1. SUFF.

2. h = p1*p2*....*pn > (h/2)^n ==>
h > (h/2)^n ==> 2^n > h^(n-1) ==> h < 2^(n/(n-1)) ==> h < 2^(1+1/(n-1)) ==> h < 4. 2,3 are prime numbers. SUFF.
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 [#permalink] New post 29 Dec 2007, 06:28

from 1, if each prime factor is greather than sq root h then only possible facor is h as it will have 1 more pime facto less than sqrt(h) otherwise so h is prime

if each prime factor is more than h/2 then also only possible factor is h so h is prime

hence any of the statement is possible to answer the question

so D is the correct choice.
  [#permalink] 29 Dec 2007, 06:28
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