if a | c and b | c does ab | c

[restin84]

I am doing a proof for a cryptography course. In part of the proof I am stating that when a and b are prime, if a | c and b | c then ab | c. Am I correct in saying this?

 

[mmm4444bot]

when a and b are prime, if a|c and b|c then ab|c

Yes, but why do a and b need to be prime?


EG:

a = 4
b = 9
c = 72

 

[lookagain]

Yes, but why do a and b need to be prime?


EG:

a = 4
b = 9
c = 72

restin84,

here is a counterexample to yours.

Let a = 2 . . . . . (prime)

Let b = 2 . . . . . . (prime)

Let c = 2


2|2 and 2|2,

but [(2)(2)] does not divide 2.

-----------------------------------------------


mmm4444bot,

your example works, but it is not true in general.


Example:
-----------

Let a = 4

Let b = 6

Let c = 12


4|12


6|12, but


[(4)(6)] does not divide 12.


--------------------------------------------------------


\(\displaystyle \text{Edit:}\)


\(\displaystyle \text{What if the revised question were if a and b are }\) distinct \(\displaystyle \text{prime numbers,}\)

\(\displaystyle \text{and c is a composite number, such that }\)

a|c and b|c, then does (ab)|c?

 

[restin84]

\(\displaystyle \text{Edit:}\)


\(\displaystyle \text{What if the revised question were if a and b are }\) distinct \(\displaystyle \text{prime numbers,}\)

\(\displaystyle \text{and c is a composite number, such that }\)

a|c and b|c, then does (ab)|c?

I did mean to state that the numbers were distinct. If I were to attempt to prove that if a|c and b|c then ab|c when a and b are distinct primes, should I use the fact that a prime number other than 2 has the form 2k+1 because it is odd? I would have two cases. One where one of the primes is 2 and another where neither of the primes are 2. Just not sure where I would start.

 

[mmm4444bot]

mmm4444bot,

your example works, but it is not true in general.

Oops. I posted that example to show that two numbers do not need to be prime to divide a third. I think that I misread the "when" statement.

 

[SAMUELK]

I am doing a proof for a cryptography course. In part of the proof I am stating that when a and b are prime, if a | c and b | c then ab | c. Am I correct in saying this?

I think the theorem states that if a and b are RELATIVELY prime, if a | c and b | c then ab | c.
(If a and b are primes, and not equal, they qualify, so your claim is correct, too.)

Ex: 6 | 18 and 9 | 18, but 54 does not. Of course, gcd(6,9) is not 1.

But 4 | 36 and 9 | 36 (as noted) above, and gcd(4,9) = 1, so 4*9 = 36 | 36.