Prime factor problem

[LS]

If h(n) = the product of all even numbers from 2 to n inclusive, what is the smallest prime factor of h(100)+1 ?

a. between 2 and 10
b. between 11 and 20
c. between 21 and 30
d. between 31 and 40
e. greater than 40


The correct answer is e. greater than 40, but why?

Thanks in advance!

 

[stapel]

I don't know if this might help, but, letting n = 2m, you can do the following:

. . .h(100) = (2)(4)(6)(8)...(n)

. . . . .= (2)(4)(6)(8)...(2m)

. . . . .= 2^m[(1)(2)(3)(4)...(m)]

. . . . .= 2^m m!

Then:

. . . . .h(100) + 1 = 2⁵⁰ 50! + 1

Can 2 be a factor? (If you divide an even number, a number for which 2 is a factor, by 2, you get a zero remainder. If you divide h(100) + 1 by 2, what will you get?)

Can 3 be a factor? (Think about the factorial. Can you divide h(100) + 1 evenly by 3?)

Continue until you see the reasoning and pattern.

Eliz.

 

[Deleted member 4993]

If h(n) = the product of all even numbers from 2 to n inclusive, what is the smallest prime factor of h(100)+1 ?

a. between 2 and 10
b. between 11 and 20
c. between 21 and 30
d. between 31 and 40
e. greater than 40


The correct answer is e. greater than 40, but why?

Thanks in advance!

This is a problem of advanced algebra - you need to use Wilson's theorem. For further discussion go to:

http://answers.yahoo.com/question/index ... 150AAzryIG

 

[LS]

Thanks for the replies, but I actually figured it out...

Given any factor f > 1 of any positive integer n >= 2, f can only also be a factor of n + m, where m is is a multiple of n. Therefore f cannot be a factor of n + 1, because 1 cannot be a multiple of n.

Since, as Stapel said, h(100) = 2^50 x (1 x 2 x 3 x ... x 50), 1 through 50 must be factors of h(100). Therefore 1 through 50 cannot be factors of h(100) + 1.

Because of this reasoning, the smallest prime factor must be greater than 50.

 

[galactus]

The smallest prime factor is 79.

I assume you knew that, though.

 

[LS]

(I don't think Wilson's is needed here, because we're not looking for a specific prime...only a general range.)

If h(n) = the product of all even numbers from 2 to n inclusive, what is the smallest prime factor of h(100)+1 ?

a. between 2 and 10
b. between 11 and 20
c. between 21 and 30
d. between 31 and 40
e. greater than 40


The correct answer is e. greater than 40, but why?

Thanks in advance!

This is a problem of advanced algebra - you need to use Wilson's theorem. For further discussion go to:

http://answers.yahoo.com/question/index ... 150AAzryIG

 

[Deleted member 4993]

(I don't think Wilson's is needed here, because we're not looking for a specific prime...only a general range.)

You are correct.

I generally stay away from prime factors and mod algebra problems - I should have done that this time too!