Manhattan GMAT Challenge Problem of the Week – 23 Jan 2012
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Question
If x is positive and not equal to 1, then the product of
for all positive integers n such that 21 ≤ n ≤ 30 is between
A. 1 and
B.and
C.and
D.and
E.and
Answer
Let’s list out the product, using “dot-dot-dot” for missing parts in the middle.
Product = (
) (
) (
)…(
)
Since the bases are all the same, we just add the exponents to get the final exponent.
Exponent of product = 1/21 + 1/22 + 1/23 + … + 1/30 (ten terms in the sum)
Now, we are not asked to compute the exact value of this number (a brutal task that would require us to find a common multiple of every integer from 21 to 30!). Rather, we want to know what the product lies between.
One way to find simple upper and lower bounds is to replace all the different fractions with the same fraction that is definitely larger or smaller. Then compute these “fake” sums and see what you get.
Start at the small end. The real sum is definitely greater than this fake sum:
1/30 + 1/30 + 1/30 + … + 1/30
We’ve replaced 1/21, 1/22, etc. with 1/30 in each position. 1/30 is definitely less than 1/21, 1/22, etc. through 1/29, so the sum is definitely less.
Lower fake sum = 1/30 + 1/30 + 1/30 + … + 1/30 (ten terms) = 10/30 = 1/3
We have a good lower bound. Let’s set an upper bound by replacing every fraction in the real sum with 1/20, which is definitely larger than every fraction in the real sum.
Upper fake sum = 1/20 + 1/20 + 1/20 + … + 1/20 (ten terms) = 10/20 = 1/2
We now have bounds on either end. The exponent must be between 1/3 and 1/2, so the product in question must be between and .
The correct answer is C.
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2 comments
Somsubhra Mukherjee on January 28th, 2012 at 5:55 pm
Fantastic problrm of exponents.We want more such examples.
Alyse on February 5th, 2012 at 10:51 pm
Exactly, so succinct and simple once you read the solution.