[superprismatic]

Let N be any positive integer. Is it true that there is always at least one

number in the set {N,N+1,N+2,N+3,N+4} which is relatively prime to the

product of the other four?

[Guest]

{1.2,2.2,3.2,4.2,5.2}

It is imposible for any decimals to be prime.

{3560,3561,3562,3563,3564,3565}

http://en.wikipedia.org/wiki/List_of_prime_numbers

[Guest]

N has to be an integer...

And it asks only for relatively prime, which means no common factors...not that the numbers can't be prime

And 3563 is relatively prime to the others (it is 7*509, which means no factors within 5)

Also, it requires either a counter-example or a number-free mathematical proof...

Something along the lines of (N+1)/N etc.

Edited February 23, 2011 by Shadow7

[Guest]

{1.2,2.2,3.2,4.2,5.2}

It is imposible for any decimals to be prime.

{3560,3561,3562,3563,3564,3565}

http://en.wikipedia.org/wiki/List_of_prime_numbers

Jetdron, N is any positive INTEGER.

[Guest]

Below are the five possible remainders when the product of four of the numbers in the set are divided by the fifth. The four numbers are shown as a product, and the remainder is after the colon:

N(N+1)(N+2)(N+3): 24/(N+4)

N(N+1)(N+2)(N+4): -6/(N+3)

N(N+1)(N+3)(N+4): 4/(N+2)

N(N+2)(N+3)(N+4): -6/(N+1)

(N+1)(N+2)(N+3)(N+4): 24/N

For the statement in the problem to be WRONG, there must be an integer N such that all five of these remainders are also integers. If such a number exists, we have found a counterexample, since all possible divisions in this counterexample will result in integers, making none of the numbers in the set relatively prime to the product of the others.

So, is there a number N such that all the above remainders are integers?

1 doesn't work because 24/(1+4) is not an integer.

2 doesn't work because -6/(2+3) is not an integer.

3 doesn't work because 24/(3+4) is not an integer.

And any number greater than 3 doesn't work because -6/(N+anything greater than 3) is not an integer.

Therefore, there is NO integer N such that all above remainders are integers.

Therefore, the problem statement is proven correct.

[Guest]

I'm not seeing how you're getting actual numbers for the numerators...and they don't have to yield integers, just simplifiable fractions...

[curr3nt]

Consider the patterns of N...N+4 with the prime factors 2 and 3. None of the numbers can share a higher prime factor. In each pattern there is at least one number that will not share a prime factor with the other four.

Factors                                                               

N   : 2 3     N   : 2       N   : 2       N   :   3     N   :         N   :         

N+1 :         N+1 :   3     N+1 :         N+1 : 2       N+1 : 2 3     N+1 : 2       

N+2 : 2       N+2 : 2       N+2 : 2 3     N+2 :         N+2 :         N+2 :   3     

N+3 :   3     N+3 :         N+3 :         N+3 : 2 3     N+3 : 2       N+3 : 2       

N+4 : 2       N+4 : 2 3     N+4 : 2       N+4 :         N+4 :   3     N+4 :         


Relatively Prime #                                                        

N+1           N+3           N+1 & N+3     N+2 & N+4     N & N+2       N, N+2 & N+4  

[superprismatic]

Consider the patterns of N...N+4 with the prime factors 2 and 3. None of the numbers can share a higher prime factor. In each pattern there is at least one number that will not share a prime factor with the other four.

Factors
N : 2 3 N : 2 N : 2 N : 3 N : N :
N+1 : N+1 : 3 N+1 : N+1 : 2 N+1 : 2 3 N+1 : 2
N+2 : 2 N+2 : 2 N+2 : 2 3 N+2 : N+2 : N+2 : 3
N+3 : 3 N+3 : N+3 : N+3 : 2 3 N+3 : 2 N+3 : 2
N+4 : 2 N+4 : 2 3 N+4 : 2 N+4 : N+4 : 3 N+4 :

Relatively Prime #
N+1 N+3 N+1 & N+3 N+2 & N+4 N & N+2 N, N+2 & N+4

You have it. Good job! Your first two sentences had the critical observation.