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Re: PS: Number property [#permalink]
23 Aug 2009, 07:27

1

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I do not think you have to calculate a lot here. The rule is that the fraction terminates if the denominator has ONLY 2 or/and 5 in its prime factorization.

A 10/189...no common factors can be canceled out, and 189 is divisible by 3. The denominator does not have ONLY 2's and 5's since it has 3 in it as well. You can stop here, concluding the fraction does not terminate... B 15/196 - no common factors can be canceled out. 196=2*2*7*7...there is a prime number 7...so the fraction does not terminate.. C 16/225...no common factors can be caceled out. decomposing the denominator 225=25*9=5^2*3^2. Does not terminate since it has 3 in its denominator D 25/144 no common factors can be canceled out. 144= 2*72, and 72 is divisible by 3...so the fraction has 3 in its denominator...does not terminate. E you do not have to check. you know it should be E since it is your last option. But if you want to be sure decompose the denominator and you will get only 2's in it

Re: PS: Number property [#permalink]
07 Sep 2010, 01:37

E. Great question. Be suspicious about 3,7 (more suspicious if the numerator has a factor of 3 or 7). Do not blindly select the answer if the denominator alone has a 3 or 7. If it is 8,4,2,6 then they are usually terminating.

Re: PS: Number property [#permalink]
07 Sep 2010, 04:30

2

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Expert's post

TriColor wrote:

Please, explain your answer. Thank you, -----------------------------------------

Q15: Which of the following fractions has a decimal equivalent that is a terminating decimal?

A. 10/189 B. 15/196 C. 16/225 D. 25/144 E. 39/128

THEORY:

Reduced fraction \frac{a}{b} (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and onlyb (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.)

the necessary and sufficient condition of terminating decimals is that the denominator's prime factors should only be 2 or 5 or both (form 2^x * 5^y)
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