Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Can you answer this conjecture? I have plenty of evidence to suggest it is correct.?
a, b, and k are positive integers. k is a prime number (not including 1). 0<a<k and 0<b<k. Prove that k is not a factor of ab.
Tomp gives an interesting answer, which I am studying. It will take me time to go through it.
However, one flaw might be that a or b could be prime numbers, though each less than k. I do not know if the proof assumes that a and or b are composite.
Perhaps, if that is the case, the proof may not work.
2 Answers
- TompLv 71 decade agoFavorite Answer
Let the prime factorisation of a be
a = p1^m1 * p2^m2 * ...... * pr^mr
where pi is one of the prime factors of a raised to the power of mi and pi <= a < k
Similarly for b
b = q1^n1 * q2^n2 * ...... * qs^ns
where qj is a prime factor of b raised to the power of nj and qj <=b < k
Note that the above two factorisations are unique to a and b.
Now
ab = (p1^m1 * p2^m2 * ...... * pr^mr)(q1^n1 * q2^n2 * ...... * qs^ns)
Since k is prime, none of the prime factors of a or b can divide k.
Conversely k cannot divide pi (1 <= i <= r) nor qj (1 <= j <= s)
Hence k cannot be a factor of the product ab
By the way, 1 is not a prime number. A prime number k is defined by the fact that it can only have two divisors, 1 and k itself.. The number 1 has only one divisor, 1 itself.
- lienad14Lv 61 decade ago
the only POSITIVE PRIME is 2 Therefore k = 2.
If a and b are positive and less than 2 then they do not exist