Number properties ...

by **iil-london** Sun Mar 23, 2008 2:47 pm

I came across the following question when completing my practice test on the GMATPrep software.
Apologies in advance, if this has already been posted ...

For every even integer 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) above 40

Would be grateful if you could share your approaches and logic used in answering this question.

Thanks in advance.

by **RonPurewal** Mon Mar 24, 2008 4:33 am

as with most problems involving numbers that are just too big to calculate, this one is about RECOGNIZING PATTERNS and deducing relevant facts from those patterns.

the number p is (2)(4)(6)(8)(10)...(98)(100) + 1.
take a look at the left-hand term of that expression (the one that contains everything except the 'plus one' part). note that ALL numbers from 1 through 50 are factors of this term:
1 is a factor, because 2 appears
2 is a factor, because 4 appears
3 is a factor, because 6 appears
4 is a factor, because 8 appears
etc.

...but, if all those numbers are factors of the left-hand term, then NONE of them (except 1) can be factors of the whole expression, because none of them goes into 1.

this means that no number between 2 and 50 (whether prime or not) can be a factor of p.

so the answer is e.

by **iil-london** Mon Mar 24, 2008 5:03 am

Thanks. What level would you rate this type of question ? 500-600 ; 600-700 ; 700+ ?

by **StaceyKoprince** Tue Mar 25, 2008 1:15 am

I'd guess 700+

Also, try searching for the text of a problem before you post - if it has already been answered, you can read the explanation right away instead of having to wait for an answer!