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A googol is the number that is written as 1 followed by 100 zeros. if

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A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post 08 May 2018, 04:18
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E

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  95% (hard)

Question Stats:

38% (02:08) correct 62% (02:04) wrong based on 107 sessions

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A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post Updated on: 08 May 2018, 05:03
1
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Solution



Given:
• Googol is the number written as 1 followed by 100 zeros
• It is represented by G

To find:
• Sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\)

Approach and Working:
• The quotients in the divisions are as follows:
    o When G is divided by 8, quotient will be 125 followed by zeros
    o When G is divided by 5, quotient will be 200 followed by zeros
    o When G is divided by 4, quotient will be 250 followed by zeros
    o When G is divided by 2, quotient will be 500 followed by zeros
    o The final sum = 125 + 200 + 250 +500 = 1075
• Therefore, the sum of the digits of the results = 1+0+7+5

As per the options, the correct answer is option A.

Answer: A
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Originally posted by EgmatQuantExpert on 08 May 2018, 04:43.
Last edited by EgmatQuantExpert on 08 May 2018, 05:03, edited 2 times in total.
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Re: A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post 08 May 2018, 04:45
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Bunuel wrote:
A googol is the number that is written as 1 followed by 100 zeros. if G represents a googol, what is the sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) ?

A. 13
B. 22
C. 107
D. 400
E. 1075


\(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) \(= 1.075G\)

Hence, \(1+7+5=13\).

Answer: A
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Re: A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post 09 May 2018, 16:36
Bunuel wrote:
A googol is the number that is written as 1 followed by 100 zeros. if G represents a googol, what is the sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) ?

A. 13
B. 22
C. 107
D. 400
E. 1075


We can get common denominators in our expression and we have:

5G/40 + 8G/40 + 10G/40 + 20G/40

43G/40

43/40 x G

1.075 x 10^100

1,075 x 10^97

We can see that the number is 1,075 followed by 97 zeros; thus, the sum of the digits is 1 + 7 + 5 = 13 (we don’t need to add any digits that are 0).

Answer: A
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Re: A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post 19 Aug 2019, 08:40
Bunuel wrote:
A googol is the number that is written as 1 followed by 100 zeros. if G represents a googol, what is the sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) ?

A. 13
B. 22
C. 107
D. 400
E. 1075


\(G=10^{100}\)
\(\frac{G}{8}\)\(=10^{100}/2^{3}=10^{97}•5^3\)
\(\frac{G}{5}\)\(=10^{100}/5=10^{99}•2\)
\(\frac{G}{4}\)\(=10^{100}/2^{2}=10^{98}•5^2\)
\(\frac{G}{2}\)\(=10^{100}/2=10^{99}•5\)
\(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}=…\)
\(10^{97}•5^3+10^{99}•2+10^{98}•5^2+10^{99}•5…=\)
\(10^{97}(5^3+10^{2}•2+10•5^2+10^{2}•5)=…\)
\(10^{97}(125+200+250+500)=…\)
\(10^{97}(1075)=…\)
\(digits:1+7+5=13\)

Answer (A).
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A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post 19 Aug 2019, 10:01
Bunuel wrote:
A googol is the number that is written as 1 followed by 100 zeros. if G represents a googol, what is the sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) ?

A. 13
B. 22
C. 107
D. 400
E. 1075


Given: A googol is the number that is written as 1 followed by 100 zeros.

Asked: if G represents a googol, what is the sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) ?

\(G= 10^100\)
\(\frac{G}{8}= \frac{1000}{8} * 10^{97} = 125 * 10^{97}\)
\(\frac{G}{5} = \frac{1000}{5}* 10^{99} = 200 * 10^{97}\)
\(\frac{G}{4} = \frac{1000}{4} * 10^{98} = 250 * 10^{97}\)
\(\frac{G}{2} = \frac{1000}{2} * 10^{99} = 500 * 10^{97}\)
\(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2} = 1075 * 100^{97}\)

Sum of the digits of \(\frac{G}{8} + \frac{G}{5} + \frac{G}{4} + \frac{G}{2}\) = 1 + 0 + 7 + 5 = 13

IMO A
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Re: A googol is the number that is written as 1 followed by 100 zeros. if  [#permalink]

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New post 22 Aug 2019, 08:16
It's simple,

write everything in powers of 10^98 as 100 is almost divided by all the denominators
Now, simplify,

you will get 107.5* 10^98
= 1075 * 10^97

This is 1075 followed by 97 zeroes

Upon adding all the digits you will get 13
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Re: A googol is the number that is written as 1 followed by 100 zeros. if   [#permalink] 22 Aug 2019, 08:16
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