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# If X = 625,000, how many nonzero digits are in the decimal representat

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Joined: 02 Sep 2009
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If X = 625,000, how many nonzero digits are in the decimal representat  [#permalink]

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29 Mar 2018, 23:53
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If X = 625,000, how many nonzero digits are in the decimal representation of 1/X ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

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Re: If X = 625,000, how many nonzero digits are in the decimal representat  [#permalink]

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30 Mar 2018, 00:01
Bunuel wrote:
If X = 625,000, how many nonzero digits are in the decimal representation of 1/X ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

1/625000 = 1/1000 x 1/625 = 0.001 x 0.0016

2 non zero digits
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Re: If X = 625,000, how many nonzero digits are in the decimal representat  [#permalink]

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31 Mar 2018, 11:18
Bunuel wrote:
If X = 625,000, how many nonzero digits are in the decimal representation of 1/X ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

I have a tendency to do these calculations mentally.
Here is how I solved it mentally-

625,000= $$5^4*1000$$
1000 wont effect the non zero digits.
1/625 will have the same digits as 1/625,000
$$1/5^4$$=$$0.2^4$$=0.0016
Hence answer is 2 non zero digits- B
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If X = 625,000, how many nonzero digits are in the decimal representat  [#permalink]

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31 Mar 2018, 11:18
Bunuel wrote:
If X = 625,000, how many nonzero digits are in the decimal representation of 1/X ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

One easy method: multiply both numerator and denominator by $$\frac{a^{b}}{a^{b}}$$ to create only powers of 10 in the denominator.
Division by 10 changes the decimal position, not the number of non-zero digits in the numerator.

Find the prime factors of 625,000
$$625,000 = (625 * 1,000) =$$
$$(5^4*10^3)=(5^4*5^32^3)$$

$$\frac{1}{X}=\frac{1}{625,000}=\frac{1}{(5^4*5^32^3)}$$

There are seven powers of 5, but only three powers of 2. We want only powers of 10 in the denominator. We need four more 2s

$$\frac{1}{(5^45^32^3)}*(\frac{2^4}{2^4})=(\frac{2^4}{10^7})$$
$$2^4 = 16$$

There will be TWO non-zero digits in the decimal representation.

(The number is .0000016)
Edit: Digvijay101 , simultaneous post. No wonder my browser got stuck.
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If X = 625,000, how many nonzero digits are in the decimal representat &nbs [#permalink] 31 Mar 2018, 11:18
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