Bunuel wrote:
A number is called “terminating in base N” if that number can be expressed as A/N^B for some integers A and B. What is the smallest positive integer N for which 5/24 is terminating in base N?
A. 1
B. 2
C. 6
D. 8
E. 24
We must find a fraction that's EQUIVALENT to 5/24 so that the EQUIVALENT fraction can be written in the form A/(N^B)
24 = (2)(2)(2)(3)
So, the denominator of the EQUIVALENT fraction must have at least three 2's
Notice that if we take 24 = (2)(2)(2)(3) and multiply both sides by
9, we get: 24(
9) = (2)(2)(2)(3)(
9)
Rewrite as: 216 = (2)(2)(2)(3)(
3)(
3)
Rewrite as: 216 = [(2)(3)][(2)(3)][(2)(3)]
Rewrite as: 216 = [(2)(3)]³
Rewrite as: 216 = 6³
This means we can take: 5/24
And multiply top and bottom by
9 to get the EQUIVALENT fraction: 45/216
Now rewrite the denominator as follows: 45/6³
In other words, 5/24 = 45/216 = 45/6³
Since 45/
6³ is written in the form A/(
N^B), we can see that
N = 6Answer: C
Cheers,
Brent
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