If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze

Question

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

Bunuel · Accepted Answer

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.

Answer: B.

jatinsachani · Answer

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.

Answer B

JeffTargetTestPrep · Answer

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.

Answer is B.

Puneethrao · Answer

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

Bunuel · Answer

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.

theGame001 · Answer

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

Bunuel · Answer

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?

WinterIsComing · Answer

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.

Bunuel · Answer

We need to multiply by 2^4/2^4 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: https://gmatclub.com/forum/if-t-1-2-9-5- ... 29447.html https://gmatclub.com/forum/if-d-1-2-3-5- ... 28457.html Hope this helps.

Abhishek009 · Answer

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 )

EMPOWERMathExpert · Answer

We are given:

d=\frac{1}{(2^3	imes 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 	imes 5^7 )} 	imes \frac{2^4}{2^4} =\frac{2^4}{(2^7*5^7 )}  
  d=\frac{2^4}{(2	imes 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  B .

BrentGMATPrepNow · Answer

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

Answer: B

Cheers, 
Brent

Kinshook · Answer

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

d=\frac{1}{2^3*5^7} = \frac{2^4}{2^75^7} = 2^4 * 10^{-7} = 16*10^{-7} = 0.0000016

2 non-zero digit (16)

IMO B

Thib33600 · Answer

Why do we choose 2^4 ?

Bunuel · Answer

In the denominator we have 2^3*5^7. If we multiply 2^3*5^7 by 2^4 we get 2^7*5^7 = 10^7. So, we need to multiply the entire fraction by 2^4/2^4 in order to convert the denominator to the base of 10 and then to convert the fraction into the decimal form: 0.xxxx.

Anonymous · Answer

Hello GMAT Experts!  I'd like your help to spot my mistake, I solved it as follows:

1/(2^3*5^7) = 1/(2^3 * 5^3 * 5^4) = 1/(10^3 * 5^4) = 1/10^3 * 1/625 so there are 3 non-zero digits.

What am I missing  JeffTargetTestPrep   Bunuel   KarishmaB   GMATCoachBen  ?

Bunuel · Answer

If it were 625*1/10^3, then this would equal 0.625 and you'd be correct. However, we have 1/625*1/10^3 and 1/625 = 0.0016, hence 1/625*1/10^3 = 0.0000016, and this has only two nonzero digits.

Anonymous · Answer

No way.. such a silly mistake! Thanks Bunuel

kokus · Answer

1/8*5 = 1/40
1/8*25=1/200
1/8*125=1/1000

The denominator increases by 5x each time we add an additional power to the base of 5. Therefore:

1/8*...=1/5000
1/8*...=1/25,000
1/8*...=1/125,000
1/8*...=1/625,000

1/625,000 -> 1/625 = 0.0016 (number of unique digits will be the same as we add/remove zeroes).

If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze

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If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze

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