How many times do you have to square 2 for it to become a multiple of 3? Is it possible? In algebraic terms, 2^x=3y.

May 29, 2017

There is no integer $x$ such that ${2}^{x}$ is divisible by $3$

Explanation:

In a similar way to classifying numbers as odd or even, we can classify numbers as one of the following:

• A multiple of $3$. That is of form $3 k$

• One more than a multiple of $3$. That is of form $3 k + 1$

• Two more than a multiple of $3$. That is of form $3 k + 2$

If we multiply a number of the form $3 k + 1$ by $2$ then we get a number of the form $3 k + 2$:

$2 \left(3 k + 1\right) = 3 \left(2 k\right) + 2$

If we multiply a number of the form $3 k + 2$ by $2$ then we get a number of the form $3 k + 1$:

$2 \left(3 k + 2\right) = 6 k + 4 = 6 k + 3 + 1 = 3 \left(2 k + 1\right) + 1$

Hence we find that successive powers of $2$ alternate between numbers of the form $3 k + 1$ and $3 k + 2$ and never $3 k$...

${2}^{0} = 1 = 3 \left(0\right) + 1$

${2}^{1} = 2 = 3 \left(0\right) + 2$

${2}^{2} = 4 = 3 \left(1\right) + 1$

${2}^{3} = 8 = 3 \left(2\right) + 2$

etc.