A number multiplied by <span>13</span> only produces an irrational number if it is itself an irrational number.
Explanation:
Suppose there were some rational number which when multiplied by <span>13</span> generated an irrational number.
If the number is rational then, by definition, it can be expressed as <span>ab</span> for some pair of integers a and b.
<span><span>ab</span>×<span>13</span>=<span><span>a×1</span><span>b×3</span></span>=<span>a<span>3b</span></span></span>
Since a is an integer and
since if b is an integer then <span>3b</span> is also an integer,
then
by definition <span><span>a3</span>b</span> is a rational number, contrary to our supposition that some rational number <span>ab</span> when multiplied by <span>13</span> could generate an irrational number.
