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# Magoosh Brain Twister: Variable Cloudiness – Answer and Explanation

Take a moment to refresh your memory, then jump into the explanation.

## Question

If a – j are distinct positive integers, what is the units digit of the lowest possible value of a^b x c^d x e^f x g^h x i^j?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 8

The first thing to notice here are the phrase “lowest possible value” and the word “distinct”. For a number to satisfy both of those conditions, it would have to be composed of the first ten positive integers, namely 1-10.

We can experiment with combinations of numbers, but we want to think logically instead of just coming up with different possible scenarios.

By pairing the highest number with ‘1’, we “neutralize” that number, since 1 to the power of anything is 1. This gives us 1^10, which equals 1. (Technically, the number doesn’t have to be ‘10’ but could be any positive integer that is not 1-9).

Next, we want to pair off the next largest number with the next smallest number. Again, we are trying to “neutralize” the effect of the large number. Be careful, though, since 9^2 is less than 2^9.

The same goes for 8^3 vs 3^8. Carrying this logic on, we get:

1^10 x 9^2 x 8^3 x 7^4 x 6^5

Don’t punch those unruly numbers into a calculator (they won’t fit on the GRE calculator). Instead, let’s figure out what the units digit for each has to be:

9^2 = 1
8^3 = 2
7^4 = 1
6^5 = 6

Multiplying those together, we get 1 x 2 x 1 x 6, which gives us 2 as the units digit.

See you again in two weeks!

### 4 Responses to Magoosh Brain Twister: Variable Cloudiness – Answer and Explanation

1. mallikarjunarao April 30, 2015 at 12:02 am #

Sir ,how does 9^2=1
8^3=2
7^4=1
6^5=6 ?

• Chris Lele May 1, 2015 at 5:07 pm #

These numbers represent the units digits:

8^3 = 51(2)
7^4 = 240(1)
6^5 = Some big number ending in a ‘6’ 🙂

Hope that clears things up!

2. Binny March 7, 2015 at 6:07 am #

Hey Chris
Nice question! I figured the integers (i.e. 1-10) and even applied the cyclicity concept yet couldn’t get to the answer because what I couldn’t guess that the integers didn’t necessarily have to be taken in order!

• Chris Lele March 9, 2015 at 11:41 am #

Hi Binny,

Glad you found the question fun! Yes, there are quite a few twists in there (the 1^10 vs the 10^1 and the fact that the integers don’t have to be in order).

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