Question

Which number produces an irrational number when multiplied by 1/3?

Answer 1

A number multiplied by 13 only produces an irrational number if it is itself an irrational number.

Explanation:

Suppose there were some rational number which when multiplied by 13 generated an irrational number.

If the number is rational then, by definition, it can be expressed as ab for some pair of integers a and b.

ab×13=a×1b×3=a3b

Since a is an integer and 
since if b is an integer then 3b is also an integer,
then
by definition a3b is a rational number, contrary to our supposition that some rational number ab when multiplied by 13 could generate an irrational number.