OG 2016: PS Question 183

[JessicaL346]

In the official guide question 183, we are told to use the last digits short-cut to solve the question. We evaluate 3^43 + 3^33 using the short-cut method. The pattern for 3 is given as 3, 9, 7, 1 repeating.

Since 40 is a multiple of 4, the units digit of 3^40 = 1. Cutting forward 3 times gives us a units digit of 3^43 to be 7.

In the answer key, it says since 32 is a multiple of 4, the units digit of 3^32 is 1. Counting forward once gives us a units digit of 3^33 to be 3.

Adding 7+3 = 10, so units digit would be 0.

My question is, for evaluating 3^33, why can't we use the fact that 30 is a multiple of 3, which gives us a units digit of 3^30 = 7. Counting forward 3 times would give us a units digit of 3^33 as 9, where 9+7 = 16 giving us a units digit of 6.

Why does this lead us to a different answer? Is it wrong to use 30 as a multiple of 3 to solve for 3^33?

Both the official guide explanation and Navigate explanation did not use this.

[RonPurewal]

hi,
please read the forum rules -- we can't use OG problems here.
that ban extends to discussion of the problems, too. (it doesn't mean "it's ok to discuss the problems without citing them", because then the ban itself would really carry no weight.)

since this is your first post, and you didn't copy the text of the problem itself, i'll let this thread stand -- but please be aware of this restriction from here onward. thanks.

__

this is one of those questions you can answer yourself, by just writing out enough examples and then looking at them for appropriate patterns.

as you noted yourself, the units digits in powers of 3 are
3, 9, 7, 1, 3, 9, 7, 1, ...
note that
...the 3rd of these is '7'
...the 6th of these is '9'
...the 9th of these is '3'
and so on.
if the kind of reasoning you're attempting to use were something that worked, then these would all be the same... but they aren't.

in fewer words—
these digits repeat in groups of FOUR, not three.
so, they won't exhibit any meaningful patterns in threes.