Divisibility proof question ?

Question

Prove the statement: For all integers a,b, and c, if ab|c then a|c and b|c.

My proof:

Suppose a,b, and c are [particular but arbitrarily chosen] integers that ab|c. [We must show that a|c and b|c]. By definition of divisibility, ab = cm for integers m. Then,

i) ab = cm

a=cm/b

therefore, a = c(m/b) (by factoring out a c)

ii) ab = cm

b = cm/a

therefore, b = c(m/a) (by factoring out a c).

So let k = m/b and q = m/a. Note that k and q are integers since they are quotient of integers where b=/= 0 and a=/=0 (by zero product property). Then,

i) a=ck

ii)b=cq.

Therefore, c divides a and c divides b by definition of divisibility.

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I just want to know if my proof is right as i feel a bit unsure.

Answer 1

if ab|c then a|c and b|c.

the above means if ab is a factor of c then a is a factor of c. But you have put it opposite that is c is a factor of ab.

But the other way is not correct

by an example

6 devides 9 * 4 but 6 does not devide 9 nor 4

facacy is becuase

a=cm/b

therefore, a = c(m/b) (by factoring out a c)

this is not correct becuase a factor of b may devide c and so on

so your proof is wrong  

Answer 2

go toohttp://www.youtube.com/watch?v=4sRYs0s9O...

Answer 3

I'm afraid you used the definition of divisibility backwards.

ab|c means there exists a k such that c = k(ab)  (c is divisible by ab)

So the proof only involves moving the brackets in this case:

c = ka(b)  Thus by the definition of divisibility b|c.

c = kb(a)  Thus a|c.

Answer 4

spot on mate