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

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In summary, the homework statement states that if a and b divide n, then a.b does not divide n. This is a contradiction.
  • #1
gottfried
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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 [itex]\frac{n}{a.b}[/itex]=I [I must not be a postitive interger]
[itex]\frac{n}{a.b}[/itex]=[itex]\frac{1}{a}[/itex].[itex]\frac{n}{b}[/itex] since a divides n it follows that n=a.x where x is a positive interger.

Therefore[itex]\frac{x}{b}[/itex]=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.
 
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  • #2
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?
 
  • #3
Is it that the products can be more than the original number 12 and therefore clearly don't divide 12.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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 ,...
 
  • #7
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.
 

1. What does it mean for a and b to divide n?

When we say that a and b divide n, it means that n is evenly divisible by both a and b. In other words, there is no remainder when n is divided by either a or b.

2. How do you prove that a.b divides n?

In order to prove that a.b divides n, we need to show that n is divisible by the product of a and b, which we can do by using the division algorithm. This algorithm states that if we have two integers a and b, there exists unique integers q and r such that n = a.b.q + r, where r is the remainder when n is divided by a.b. If r is equal to 0, then we can conclude that a.b divides n.

3. Can you provide an example of a and b dividing n?

Sure, let's say a = 4, b = 2, and n = 24. We can see that 24 is evenly divisible by both 4 and 2, as 24/4 = 6 and 24/2 = 12. Therefore, a.b = 4.2 = 8, and 8 divides 24.

4. How can you disprove that a.b divides n?

To disprove that a.b divides n, we need to find a counterexample. This means finding values for a, b, and n such that n is not divisible by a.b. For example, if a = 3, b = 5, and n = 12, we can see that 12 is not divisible by 3.5 = 15, therefore disproving the statement.

5. What is the significance of proving or disproving a.b divides n?

Proving or disproving a.b divides n is important because it helps us understand the relationship between different numbers and their divisibility. It also allows us to make conclusions about the factors of a given number, and can be useful in solving mathematical problems and proofs.

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