Question

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

Answer 1

We can simply observe that.

1. 0.777... is rational, because it is a number with infinite but repeating decimal part.
2. 1/3 is rational, because it's the division between two integers
3. $-\sqrt{16}=-4$, so this is rational as well.

Since the product of two rational numbers is always rational, we have that

$0.\bar{7}\cdot \dfrac{1}{3},\quad \dfrac{1}{3}\cdot \dfrac{1}{3},\quad -4\cdot \dfrac{1}{3}$

are all rationals, since they are the product of two rationals.

On the other hand, we have

$\sqrt{27}=\sqrt{9\cdot 3}=3\sqrt{3}$

and thus

$3\sqrt{3}\cdot \dfrac{1}{3}=\sqrt{3}$

which is irrational.