+1 vote
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for

1. given any two specific numbers

2. any two arbitrary numbers

If we can have Turing enumerator which can generate strings in lexicographic order , then the set is known as recusive(decidable) one else if order is not maintained but Turing enumeration is possible then it is recursively enumerable..

So in this case the problem is to compute the product of two numbers ..So we can enumerate the results in sequence and hence let Turing enumerator enumerates the string in ordered (lexicographic) manner as every number can be written in the form of product of 2 numbers..

U can also understand like this :

A problem is said to be decidable iff we have an algorithm for it..So for multiplication trivially we have algorithm which can be implemented in a computer..

So equivalently we can also have Halting Turing Machine for this purpose..Hence it is a recursive(decidable) set..

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thanks, got it

I think it is undecidable problem. You can check here.

http://gateoverflow.in/84835/gate1990-3-vii#c140267
Because the question is not asking there exist a TM which can do the product of two numbers. Instead, it is asking given any arbitrary TM, does it computes the product of two numbers.

i agree with you, may be turing machine doesnt halt on given input, so how can we say that it will compute product pf two numbers..
+1 vote
A Turing machine can compute product of any two numbers, hence decidable problem. However, if the question is asked to find out whether a given Turing machine can perform product of two numbers, then it is an undecidable problem
[ Proof: design a (halting) Turing machine which can compute product of two number (recursive & decidable). Now give the same input to the given Turing machine and compare the output with your TM's output. If you can enumerate this to all the natural number then you can say the given TM can perform the product of two number. Since the given TM is unknown to us, it could be possible that for any two number given as a input to the TM might cause it to loop infinitely and it might never halt (halting problem). If that happens, then the entire problem of whether a particular TM can perform product of two numbers will become undecidable. ]