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What is the smallest positive integer n such that n / 420 ca

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by Max@Math Revolution » Thu Mar 21, 2019 1:15 am

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[GMAT math practice question]

What is the smallest positive integer n such that n / 420 can be expressed as a terminating decimal?

A. 18
B. 21
C. 24
D. 30
E. 42
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by GMATGuruNY » Thu Mar 21, 2019 1:45 am
Max@Math Revolution wrote:[GMAT math practice question]

What is the smallest positive integer n such that n / 420 can be expressed as a terminating decimal?

A. 18
B. 21
C. 24
D. 30
E. 42
A fraction will yield a terminating decimal if the prime-factorization of its denominator contains only 2's and/or 5's.
n/420 = n/(2*2*3*5*7).
For a terminating decimal to be yielded, the prime factors in red must CANCEL OUT with n, leaving only 2's and 5's in the denominator.
Thus, the smallest possible value for n = 21:
21/420 = (3*7)/((2*2*3*5*7) = 1/(2*2*5).
Since the prime-factorization in green contains only 2's and 5's, the resulting decimal will be terminating.

The correct answer is B.
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by Max@Math Revolution » Sun Mar 24, 2019 5:04 pm
=>

n / 420 = n / {2^2*3^1*5^1*7^1}
The fraction will only be a terminating decimal if both 3 and 7 can be canceled out.
Thus, the smallest possible value of n is 3*7 = 21.

Therefore, the answer is B.
Answer: B
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