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# If h (x) is the product of the integers from 1 to x, the

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Senior Manager
Joined: 10 Dec 2004
Posts: 274
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Kudos [?]: 67 [0], given: 0

If h (x) is the product of the integers from 1 to x, the [#permalink]  10 Jan 2005, 10:05
If h (x) is the product of the integers from 1 to x, the least prime factor of h (100)+1 is in which of the following ranges?

2 to 10
10 to20
20 to 30
30 to 40
above 40

How do we solve it? I have OA.
VP
Joined: 18 Nov 2004
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Kudos [?]: 22 [0], given: 0

Wud go with 10 to 20, will explain if correct. What's the OA ?
Senior Manager
Joined: 10 Dec 2004
Posts: 274
Followers: 1

Kudos [?]: 67 [0], given: 0

10-20 is incorrect
However you may want to explain your reasons..OA might be wrong and your explanation might give some lead.

OA is above 40
Senior Manager
Joined: 10 Dec 2004
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Kudos [?]: 67 [0], given: 0

Dan you are correct. Please explain.
Director
Joined: 03 Nov 2004
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Kudos [?]: 19 [0], given: 0

Since h(100)+1 is 101, the range should be 'above 40'
Senior Manager
Joined: 10 Dec 2004
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Kudos [?]: 67 [0], given: 0

h(100)+1 cannot be 101
h(100) is product of first 100 ints.
Manager
Joined: 28 Aug 2004
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Kudos [?]: 1 [0], given: 0

Take a smaller number as an example.

5!

is divisible by all numbers up to and including 5. This includes primes 2, 3 and 5. Now, if that's true, then 5! + 1 is not divisible by any of those numbers. 1 is not prime. and 5! is not divisible by 2 since 5! (or in the question 100!) is even and 5! + 1 is odd.

so for 100! + 1, the least prime has to be greater than not only 40 but also 100.
Director
Joined: 19 Nov 2004
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Location: SF Bay Area, USA
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Very interesting problem.

h (100)+1 = 100! +1

This will not be a factor of every integers from 2 to 100, so 0...40 makes sense.

Good one. Keep it coming!
CIO
Joined: 09 Mar 2003
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nocilis wrote:
Very interesting problem.

h (100)+1 = 100! +1

This will not be a factor of every integers from 2 to 100, so 0...40 makes sense.

Good one. Keep it coming!

yup, i agree. That number will be one more than any multiple of all the numbers between 2-100, so it will be divisible by none of them.
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