THEORY:Reduced fraction
\frac{a}{b} (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal
if and only b (denominator) is of the form
2^n5^m, where
m and
n are non-negative integers. For example:
\frac{7}{250} is a terminating decimal
0.028, as
250 (denominator) equals to
2*5^3. Fraction
\frac{3}{30} is also a terminating decimal, as
\frac{3}{30}=\frac{1}{10} and denominator
10=2*5.
Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.For example
\frac{x}{2^n5^m}, (where x, n and m are integers) will always be terminating decimal.
(We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction
\frac{6}{15} has 3 as prime in denominator and we need to know if it can be reduced.)
Questions testing this concept:
does-the-decimal-equivalent-of-p-q-where-p-and-q-are-89566.htmlany-decimal-that-has-only-a-finite-number-of-nonzero-digits-101964.htmlif-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-d5-e-is-p-q-a-terminating-decimal-125789.html700-question-94641.htmlis-r-s2-is-a-terminating-decimal-91360.htmlpl-explain-89566.htmlwhich-of-the-following-fractions-88937.htmlif-r-and-s-are-positive-integers-141000.htmlBACK TO THE ORIGINAL QUESTION:If x is an integer, can the number (5/28)(3.02)(90%)(x) be represented by a finite number of non-zero decimal digits?First of all
\frac{5}{28}*3.02*\frac{9}{10}*x=\frac{5*302*9*x}{28*100*10}=\frac{5*302*9*x}{7*(4*100*10)}=\frac{5*302*9*x}{7*(2^2*2^2*5^2*2*5)}. Now, according to the theory above, in order this number to be termination decimal, 7 must be reduced by a factor of x (no other number in the numerator has 7 as a factor and all other numbers in the denominator have only 2's and 5's), so it'll be a terminating decimal if x is a multiple of 7.
(1) x is greater than 100. Not sufficient.
(2) x is divisible by 21. Sufficient.
Answer: B.
Hope it's clear.
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