definition An integer n is called a perfect cube if nk3 for

definition: An integer n is called a perfect cube if n=k^3 for some integer k. FInd the flaws in the following proof. then write a correct proof. Statement: If n,m are perfect cubes then so is mn. Proof: Let m and n be perfect cubes. Then n=k^3 and m=l^3 for all integers k,l. Then mn=k^3l^3. Therefore, mn is a perfect cube.

Solution

Proof: Let m and n be perfect cubes. Then n=k^3 and m=l^3 for all integers k,l. Then mn=k^3l^3. Therefore, mn is a perfect cube

Here the flaw is in the second sentence.

i.e.  n=k^3 and m=l^3 for all integers k,l should be worded as

n=k^3 and m=l^3 for one integer k and another integer l. (i.e. for all integers is wrong)

Next it follows that

mn = kl^3 hence mn is a perfect cube