Consecutive integers

Jun 2008
148
10
Prove that for any set of six consecutive integers (all greater than 5), at least one integer is not divisible by 2,3 or 5.
 
May 2008
2,295
1,663
Prove that for any set of six consecutive integers (all greater than 5), at least one integer is not divisible by 2,3 or 5.
Hint: look at your set modulo 6 and note that for any integer \(\displaystyle k,\) either \(\displaystyle 6k+1\) or \(\displaystyle 6k+5\) is not divisible by 5.

by the way, do we really need the condition in red?
 
Jun 2008
148
10
Oops. I guess we don't need the condition in red.

Too finish off the hint, I'd show that if 6k+1 is divisible by 5, then 6k+5 = 4mod5. Similarily if 6k+5 = 0mod5, then 6k+1 = (-4) = 1mod5. As both of these are clearly not divisible by 2 or 3, it follows that one of them is not divisible by 2,3, or 5.

Question #2:

Show that any set of 8 consecutive integers have the property that at least one member in the set is not divisible by 2,3,5 or 7.

I guess maybe for this one, we would look at things modulo 30, yes?

Question #3:

Show by counterexample that the following statement is false:

Any set of 2k consecutive integers (all greater than p_k) have the property that at least one member of the set is not divisible by the any of the primes up to p_k (inclusive).