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Any decimal that has only a finite number of nonzero digits

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Any decimal that has only a finite number of nonzero digits [#permalink]

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New post 13 Jan 2016, 19:09
lpetroski wrote:
Brunel -

Just to be 100% clear -- ANY fraction must have a denominator of 2, 5, or some multiple of the two numbers to terminate?


Yes. Fractions with denominators multiples of 2 or 5 or 2 AND 5 ONLY will give you a terminating decimal representation.

Examples. x/25, x/100, x/125, x/200 etc.

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Re: Any decimal that has only a finite number of nonzero digits [#permalink]

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New post 12 May 2016, 19:23
My approach to solve this problem:
p/q = (2^(a-c)*3^(b-d))/5^e
statement 1) a>c.
=> the 2^(a-c) is an integer, more so , since 2 is positive, it will return a positive integer.
case1) b>d then 3^(b-d) is an integer . Now whatever be the value of e (again integer) we will have a denominator with powers of 5. We know that if we divide by 5 or its powers, we always get a terminating decimal. The reason is that we are always multiplying 5 with itself. To see this logic. let us say we are dividing a number by 5^2. Simply divide the number by the first 5. This always gives a terminating decimal. Then divide that terminating decimal number with another 5 remaining in the denominator. This logic can be extended to any powers of 5.

case 2) let us try to reduce this to 2/3 ..it is possible to set b-d = 1 and e=0 and a-c=1. We know 2/3 is non terminating
Therefore insuff

statement #2
b>d:- This guarentees that 3^(b-d) is an integer.
Now if a> or =c and e>0 this implies that we will divide an integer by 5 .. answer is terminating decimal
now if a <c and e >0 this implies that we divide integer by 10, answer is yes
what if e< 0 then either we will have p/q as integer or a number divisible by 2 .. we know this will also be a terminating decminal

So statement #2 is sufficient -
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Re: Any decimal that has only a finite number of nonzero digits [#permalink]

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New post 17 May 2016, 21:11
Yes for terminating decimal denominator must be either 2 or 5 or combination of 2 & 5 (2*5)

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Re: Any decimal that has only a finite number of nonzero digits [#permalink]

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New post 25 Oct 2017, 12:05
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Re: Any decimal that has only a finite number of nonzero digits   [#permalink] 25 Oct 2017, 12:05

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