Question 1192007: Prove:
if 3 is not divisible by m * n, then 3 is not divisible by m and 3 is not divisible by n
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Proof by contradiction:
Assume that 3 is a factor of m. We'll show a contradiction arises from this assumption.
This means m = 3k for some integer k
This further leads to mn = 3k*n = 3(kn)
Showing that mn is a multiple of 3.
But this contradicts the fact that mn is not a multiple of 3.
Therefore, we must have m be a non-multiple of 3 as well.
Similar steps would apply to show that n must be a non-multiple of 3. This is one application of "without loss of generality" (WLOG) you can do.
Ultimately you should find that we get a contradiction if either m = 3k or n = 3p for integers k and p. Therefore, m and n cannot be multiples of 3.
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
Prove:
if 3 is not divisible by m * n, then 3 is not divisible by m and 3 is not divisible by n
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The posted statement is INCORRECT.
One possible counter-example is m = 5, n = 1.
It is true that 3 is not divisible by m*n = 5*1 = 5, in this case - - - so the premise is TRUE.
But the "then" statement is FALSE:
it is true that 3 is not divisible by 5,
but it is FALSE that 3 is not divisible by 1.
This counter-example DISPROVE your statement in the post.
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Solved and DISPROVED. Answered and explained.
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From the first glance on your post it is clear that it is like a sock turned inside out,
and the correct statement must be different - but in this case, I prefer do not discuss
my suspicions and instead disprove the wrong statement.
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