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How many digits are there in the integer 8^7 × 25^12?

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Bunuel
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How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   30 Apr 2016, 03:38
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How many digits are there in the integer 8^7 × 25^12?

A. 27
B. 24
C. 21
D. 19
E. 15
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B
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ronny123
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   30 Apr 2016, 23:29
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8^7 × 25^12

This can be simplified to
2^21 × 5^24 (since 2^3 = 8 and 5^2 = 25)

Now 2 x 5 =10

So, simplifying further
(2^21 × 5^21) x 5^3
= 10^21 x 5^3
= 125 x 10^21

10^21 has 1 one and 21 zeros which when multiplied with 125 gives 125 followed by 21 zeros
So, in total, 24 digits

Correct Option: B
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OptimusPrepJanielle
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   02 May 2016, 02:16
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Bunuel wrote:
How many digits are there in the integer 8^7 × 25^12?

A. 27
B. 24
C. 21
D. 19
E. 15


We can easily calculate the number of digits in teh end, if we know the number of 0's at the end.

8^7 × 25^12 = (2^3)^7*(5^2)^12 = 2^21*5^24 = 5^3*(10)^21 (Converting into the powers of 10)
= 125*10^21

Total digits = 21 0's + 3 digits = 24

Correct Option: B
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   24 May 2017, 12:38
What I did was break it down to the prime numbers - 5^24 and 2^21 and took 24 as my answer bc it was the largest exponent.

I used 5^2 * 2^1 = 20 to verify my approach, but idk if it just works in this case. Is this a valid method?

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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   24 May 2017, 13:22
Bunuel wrote:
How many digits are there in the integer 8^7 × 25^12?

A. 27
B. 24
C. 21
D. 19
E. 15


\(8^7*25^{12}\)

\((2^3)^7*(5^2)^{12}\)

\(2^{21}*5^{24}\)

\((2*5)^{21}*5*5*5\)

\(125*10^{21}\)

That's 3+21=24 digits.

B.
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   26 May 2017, 11:46
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Bunuel wrote:
How many digits are there in the integer 8^7 × 25^12?

A. 27
B. 24
C. 21
D. 19
E. 15


Let’s prime factorize the given terms:

8^7 × 25^12 = (2^3)^7 x (5^2)^12 = 2^21 x 5^24

We see that we have 21 five-by-two pairs, which create 21 trailing zeros.

We are left with 5^3 = 125, which is 3 digits.

Thus, 8^7 × 25^12 has a total of 21 + 3 = 24 digits.

Answer: B
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   16 Feb 2018, 17:14
gmathopeful19 wrote:
What I did was break it down to the prime numbers - 5^24 and 2^21 and took 24 as my answer bc it was the largest exponent.

I used 5^2 * 2^1 = 20 to verify my approach, but idk if it just works in this case. Is this a valid method?

Posted from my mobile device


This is how I did it as well. I can't tell if I got lucky or if I'm a low key genius.
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How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   17 Feb 2018, 08:47
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jsheppa wrote:
gmathopeful19 wrote:
What I did was break it down to the prime numbers - 5^24 and 2^21 and took 24 as my answer bc it was the largest exponent.

I used 5^2 * 2^1 = 20 to verify my approach, but idk if it just works in this case. Is this a valid method?

Posted from my mobile device


This is how I did it as well. I can't tell if I got lucky or if I'm a low key genius.


Hi jsheppa & gmathopeful19,

This is not a valid approach. as per this logic if I have \(2^3*3^2\), then I should have \(3\) digits but \(2^3*3^2=72\), i.e. two digits only.
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink] New post   27 Mar 2024, 09:15
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Re: How many digits are there in the integer 8^7 × 25^12? [#permalink]
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