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#.

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Answer 1 · 2017-05-29T07:31:44

There is no integer #x# such that #2^x# is divisible by #3#

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

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.