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.

1 Answer
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 3k

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

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

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

2(3k+1) = 3(2k)+2

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

2(3k+2) = 6k+4 = 6k+3+1 = 3(2k+1)+1

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

2^0 = 1 = 3(0)+1

2^1 = 2 = 3(0)+2

2^2 = 4 = 3(1)+1

2^3 = 8 = 3(2)+2

etc.