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Manager  Joined: 02 Dec 2012
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If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 57% (01:33) correct 43% (01:44) wrong based on 2961 sessions

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If $$d=\frac{1}{2^3*5^7}$$ is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten
Math Expert V
Joined: 02 Sep 2009
Posts: 62291
Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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76
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Given: $$d=\frac{1}{2^3*5^7}$$.

Multiply by $$\frac{2^4}{2^4}$$ --> $$d=\frac{2^4}{(2^3*5^7)*2^4}=\frac{2^4}{2^7*5^7}=\frac{2^4}{10^7}=\frac{16}{10^7}=0.0000016$$. Hence $$d$$ will have two non-zero digits, 16, when expressed as a decimal.

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Intern  Joined: 24 Aug 2013
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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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14
4
Another approach:

$$\frac{1}{(2^3*5^7)}$$ =$$\frac{1}{(2^3*5^3*5^4)}$$ by splitting denominator.

= $$\frac{1}{(10^3*5^4)}$$ = $$\frac{10^{-3}}{5^4}$$

Representing numerator as$$\frac{(10^4*10^{-7})}{5^4}$$ = $$2^4*10^{-7}$$ = $$16*10^{-7}$$

=.0000016 , Hence 2 digits.

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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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1/2^3*5^7 = 2^-3*5^-7 =.002 * .0000007. So there are 2 non zero digits!!
Math Expert V
Joined: 02 Sep 2009
Posts: 62291
Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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6
1
Puneethrao wrote:
1/2^3*5^7 = 2^-3*5^-7 =.002 * .0000007. So there are 2 non zero digits!!

Unfortunately this is not correct:

$$2^{-3}=\frac{1}{8}=0.125$$ not 0.002, which is 2/10^3 and $$5^{-7}=\frac{1}{78,125}=0.0000128$$ not 0.0000007, which is 7/10^7.

Hope it helps.
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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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Bunuel wrote:
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Given: $$d=\frac{1}{2^3*5^7}$$.

Multiply by $$\frac{2^4}{2^4}$$ --> $$d=\frac{2^4}{(2^3*5^7)*2^4}=\frac{2^4}{2^7*5^7}=\frac{2^4}{10^7}=\frac{16}{10^7}=0.0000016$$. Hence $$d$$ will have two non-zero digits, 16, when expressed as a decimal.

I have seen couple of more problem like this. One thing is still not clear to me. When you multiply whole denominator by 2^4 why is 5^7 getting ignored? Shouldn't 2^4 multiply both 2^3 as well as 5^7?

Thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 62291
Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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2
theGame001 wrote:
Bunuel wrote:
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Given: $$d=\frac{1}{2^3*5^7}$$.

Multiply by $$\frac{2^4}{2^4}$$ --> $$d=\frac{2^4}{(2^3*5^7)*2^4}=\frac{2^4}{2^7*5^7}=\frac{2^4}{10^7}=\frac{16}{10^7}=0.0000016$$. Hence $$d$$ will have two non-zero digits, 16, when expressed as a decimal.

I have seen couple of more problem like this. One thing is still not clear to me. When you multiply whole denominator by 2^4 why is 5^7 getting ignored? Shouldn't 2^4 multiply both 2^3 as well as 5^7?

Thanks

Frankly, the red part does not make any sense...

The denominator is $$2^7*5^7$$. Multiply it by $$2^4$$. What do you get?
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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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Bunuel wrote:
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Given: $$d=\frac{1}{2^3*5^7}$$.

Multiply by $$\frac{2^4}{2^4}$$ --> $$d=\frac{2^4}{(2^3*5^7)*2^4}=\frac{2^4}{2^7*5^7}=\frac{2^4}{10^7}=\frac{16}{10^7}=0.0000016$$. Hence $$d$$ will have two non-zero digits, 16, when expressed as a decimal.

What is it that you saw that indicated you should multiply by 2^4. Just looking at the problem that never occurred to me and I'd like to understand why it did to you.
Math Expert V
Joined: 02 Sep 2009
Posts: 62291
Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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2
1
WinterIsComing wrote:
Bunuel wrote:
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Given: $$d=\frac{1}{2^3*5^7}$$.

Multiply by $$\frac{2^4}{2^4}$$ --> $$d=\frac{2^4}{(2^3*5^7)*2^4}=\frac{2^4}{2^7*5^7}=\frac{2^4}{10^7}=\frac{16}{10^7}=0.0000016$$. Hence $$d$$ will have two non-zero digits, 16, when expressed as a decimal.

What is it that you saw that indicated you should multiply by 2^4. Just looking at the problem that never occurred to me and I'd like to understand why it did to you.

We need to multiply by 2^6/2^6 in order to convert the denominator to the base of 10 and then to convert the fraction into the decimal form: 0.xxxx.

Similar questions to practice:
if-t-1-2-9-5-3-is-expressed-as-a-terminating-decimal-ho-129447.html
if-d-1-2-3-5-7-is-expressed-as-a-terminating-decimal-128457.html

Hope this helps.
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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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9
5
If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Since actually dividing 1/(2^3*5^7) would be time consuming, we want to manipulate d so that we are working with a cleaner denominator. The easiest way to do that is to multiply d by a value that will produce a perfect power of 10 in the denominator. This means that the number of 2s in the denominator will equal the number of 5s in the denominator.

Thus, we can multiply 1/(2^3*5^7) by 2^4/2^4. This gives us:

2^4/(2^7*5^7)

2^4/10^7

16/10^7

16/10,000,000

We can stop here because we know that the 10,000,000 in the denominator means to move the decimal place after the 16 seven places to the left. The final value of d will be 0.0000016. Note that the division of 16 by 10,000,000 did not produce any additional non-zero digits. Thus d has 2 non-zero digits.

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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

$$d$$ = $$\frac{1}{(2^3*5^7)}$$

=>$$d$$ = $$\frac{1}{(2^3*5^3*5^4)}$$

=>$$d$$ = $$\frac{1}{(10^3*5^4)}$$

$$\frac{1}{5}$$ = $$0.20$$

$$\frac{1}{25}$$ = $$\frac{0.20}{5}$$ => $$0.04$$

$$\frac{1}{125}$$ = $$\frac{0.04}{5}$$ => $$0.008$$

$$\frac{1}{625}$$ = $$\frac{0.008}{5}$$ => $$0.0016$$

Hence there will be 2 non zero digits...

Feel free to revert in case of any doubt ( I have used some shortcuts , would love to explain if needed )

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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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1
3
We are given:

$$d=\frac{1}{(2^3\times 5^7 )}$$

In the denominator, we have two numbers with different bases and different exponents. We can rewrite those numbers to have same exponents.

$$d=\frac{1}{(2^3 \times 5^7 )} \times \frac{2^4}{2^4} =\frac{2^4}{(2^7*5^7 )}$$

$$d=\frac{2^4}{(2\times 5)^7}$$

$$d=\frac{2^4}{10^7}$$

$$2^4 = 16$$ and $$10^7$$ gives us $$7$$ decimal places. We can write this as:

$$0.0000016$$

We have 2 non-zero digits. The final answer is .
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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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Top Contributor
1
If $$d=\frac{1}{(2^3)(5^7)}$$ is expressed as a terminating decimal, how many nonzero digits will d have?

(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten

Let's take the fraction $$\frac{1}{(2^3)(5^7)}$$ and find an equivalent fraction that has a power of 10 in its denominator.

Why do this?
Well, it's very easy to take a fraction with a power of 10 in its denominator and convert it to a decimal

For example:
$$\frac{33}{1,000}=0.033$$

$$\frac{1891}{1,000,000}=0.001891$$

$$\frac{7}{100,000}=0.00007$$

Okay, let's begin...

Take: $$\frac{1}{(2^3)(5^7)}$$

Multiply numerator and denominator by $$2^4$$ to get: $$\frac{2^4}{(2^4)(2^3)(5^7)}$$

Simplify denominator to get: $$\frac{2^4}{(2^7)(5^7)}$$

Rewrite denominator as follows: $$\frac{2^4}{10^7}$$ ASIDE: I applied the rule that says $$(x^k)(y^k)=(xy)^k$$

Simplify: $$\frac{16}{10,000,000}=0.0000016$$

Cheers,
Brent
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If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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Alternate solution:

Focus on the denominator:

$$2{^3}*5{^7}= 2{^3}*5{^3}*5{^4}$$
We need $$2{^4}$$ to get $$10{^7}$$ which will end the terminating decimal but since we are short $$2{^4}$$ hence, we will have the last digit as $$2{^4}$$ = 16 therefore 2 non zero digits!

Originally posted by Kritisood on 18 Feb 2020, 09:05.
Last edited by Kritisood on 23 Mar 2020, 02:39, edited 1 time in total.
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Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze  [#permalink]

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First of all we can notice that is possible to obtain 1/10^3 * 1/5^4 and we know that 5^4 = 625.  Being asked just the number of non zero digit we can also not consider 1/10^3 and just focalize on the second term.  To make the division of this term on paper we have to multiply the numerator for some power of ten (10^3) and divide it for 625. The quotient that we will obtain should be have the same number of zero as the one that we have added to 1, but in this case this consideration is irrelevant.  1000:625 = 1,5.  So the answer will be B   [Answer B] Re: If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze   [#permalink] 23 Mar 2020, 02:16
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