Generating non-primes
- Give me three distinct primes?
- Will 5, 11, and 19 do?
- That’s fine.
- What is the sum of the sum and the product of these?
- Did you swallow a whale?
- What is 5 + 11 + 19 + 5 * 11 * 19?
- 1080.
- Prove that regardless of which three distinct primes you start with, p + q + r + p * q * r is never a prime.
- Not even once?
Problem source: Berkeley Math Circle.
June 24th, 2010 at 12:20 am
I feel like I must be missing something, but …
Assume that p, q and r are distinct odd primes. Then p + q + r is odd, and pqr is also odd. So p+q+r + pqr is even (and greater than 2). It can’t be prime.
Suppose one of p, q, r is 2 and the others are distinct odd primes. Then p + q + r is even and pqr is even as well, so p+q+r + pqr is even (and greater than 2). It can’t be prime.
June 24th, 2010 at 2:55 am
Why missing something?
It’s nice how easy it turns out to be, isn’t it, when at first glance it looks like it will be tough?
June 24th, 2010 at 8:22 am
Funny. Lovely little gem.
All primes end with 1, 2, 3, 5, 7, or 9.
That gives the following options:
1, 2, 3
1, 2, 5
1, 2, 7
1, 2, 9
1, 3, 5
1, 3, 7
1, 3, 9
1, 5, 7
1, 5, 9
1, 7, 9
2, 3, 5
2, 3, 7
2, 3, 9
2, 5, 7
2, 5, 9
3, 5, 7
3, 5, 9
5, 7, 9
Combine each of those combinations and you end up with an even number. Only one even number is prime and needless to say, none of these = 2.