Take a moment to refresh your memory, then jump into the explanation.
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?
Answer and Explanation
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!