If a and b divide n prove or disprove that a.b divides n.

[gottfried]

Homework Statement

If a and b divide n prove or disprove that a.b divides n.[a,b,n are positive intergers]

Homework Equations

The Attempt at a Solution

Suppose a.b does not divide n.
Then $\frac{n}{a.b}$=I [I must not be a postitive interger]
$\frac{n}{a.b}$=$\frac{1}{a}$.$\frac{n}{b}$ since a divides n it follows that n=a.x where x is a positive interger.

Therefore$\frac{x}{b}$=I
This implies that the quotient of two positive intergers cannot be an interger and this is a contradiction.

I have a feeling this proof is not sufficient since it is a only a contradiction in certain cases where b divides x. If anybody could tell me if this proof is sufficient and/or give me a better one that would be appreciated.

 

[Bacle2]

Think of a number with a lot of divisors, say 12 :

Divisors are : 2,3,4,6, 12...

Can you spot an issue with the products?

 

[gottfried]

Is it that the products can be more than the original number 12 and therefore clearly don't divide 12.

 

[Bacle2]

Right. 4|12, and 6|12 , but (4)(6)=24. Your statement, (I think) , would be true,if

you restricted yourself to divisors of n smaller than Sqr(n) , the square root of n.

 

[gottfried]

That does makes sense that would work and I also think it would work if the a and b were relative primes such that (a,b)=1.

 

[Bacle2]

You're right, 6|300 and 15|300, but 90 does not. This is also a counter to the

original of a different type--since 90<300 ,so that it could in theory divide it.

Maybe this counterargument would also work:

If a|n and b|n implied ab|n without qualification,then we could extend this to :

a|n and ab|n would imply a(ab)|n . Then a(ab)|n and b|n would imply a(ab)b|n ,...

 

[Arian.D]

however, if the G.C.D of a and b is 1, then ab does divide n.
In general, the L.C.M of a and b does divide n by definition.