Bunuel wrote:
Which of the following does NOT have a decimal equivalent that is a terminating decimal?
A. 1/2^25
B. 5^3/2^7
C. 3^8/6^10
D. 3^9/6^7
E. 6^8/10^10
Kudos for a correct solution.
I will admit to having taken more time to solve this one--2:59--than I would like to have seen, but that is how it goes sometimes, and that just lets me know which areas I need to practice to improve my efficiency. I approached the problem in a more intuitive manner, without knowing the 2's and 5's trick, but sort of feeling out the numbers as I went along.
(A) I traced the pattern 1/4, 1/8, 1/16, and then thought that although there would be a bunch of zeros after the decimal, these even divisors would eventually lead to a terminating decimal.
(B) I actually worked the problem to 125/128, and then I thought that I might want to check other answers for more obvious tip-offs (like 1/3 or 1/9).
(C) Here, I started to work down the fractions by common factors, so this answer became \(\frac{3^8}{(2^1^0 * 3^1^0)}\), which ended up as
\(\frac{1}{(2^1^0 * 3^2)}\). That 3^2 (or 9) in the denominator looked promising...
(D) Performing the same factoring as in (C), this choice became \(\frac{3^9}{(3^7 * 2^7)}\), which reduced to \(\frac{3^2}{2^7}\), and 9 divided by an even number did not feel as promising as (C).
(E) I did not even carry out anything, since 10^10 would lead to division by 1 with a bunch of zeros.
Between (C) or (D), to be honest, the 9 of the denominator in (C) persuaded me to go for it. That is, I remembered that 1/9 is .111111, 2/9 is .222222, and so on, so I reasoned that 1 divided by 9, even if that 9 were multiplied by something else, would still lead to some sort of nonterminating decimal. Ugly, yes, but still correct.
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