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

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*1 b*3 = a 3b

Since a is an integer and

since if b is an integer then 3b is also an integer,

then

by definition a3 b is a rational number, contrary to our supposition that some rational number ab when multiplied by 13 could generate an irrational number.