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How many digits are there in the product 2^23*5^24*7^3?

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Bunuel
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How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   03 Feb 2015, 09:53
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How many digits are there in the product 2^23*5^24*7^3?

A. 24
B. 25
C. 26
D. 27
E. 28
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D
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kdatt1991
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   03 Feb 2015, 11:41
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Bunuel wrote:
How many digits are there in the product 2^23*5^24*7^3?

A. 24
B. 25
C. 26
D. 27
E. 28


Kudos for a correct solution.


Seems like a tricky question, but I hope that I have been able to crack it! Here's my solution:

\((2^{23})*(5^{24})*(7^{3}) = (2^{23})*(5^{23})*(5)*(7)*(7)*(7)\)

\((2^{23})*(5^{23})*(5)*(7)*(7)*(7) = ((2*5)^{23})*(5)*(7)*(7)*(7)\)

\(((2*5)^{23})*(5)*(7)*(7)*(7) = (10^{23})*(35)*(49)\)...... From this step onwards it is probably possible to estimate the number of digits by approximating \((10^{23})*(35)*(49)\) to \((10^{23})*(35)*(50)\)!

But, just to make sure:

\((10^{23})*(35)*(49)\) = \((10^{23})*(35)*(50-1)\).... Therefore \((10^{23})*(1750 - 35)\) which is can be simplified to \((10^{23})*(1715)\)

\((10^{23})*(1715)\) should have exactly 27 digits!

I think the answer is D!

Please consider giving me KUDOS if you felt this post was helpful and correct! or please enlighten me (in case my answer's incorrect) so that I can learn and improve from my mistakes! Thanks. :)
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   09 Feb 2015, 04:57
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Bunuel wrote:
How many digits are there in the product 2^23*5^24*7^3?

A. 24
B. 25
C. 26
D. 27
E. 28


Kudos for a correct solution.


VERITAS PREP OFFICIAL SOLUTION:

The key to this problem is rearranging the math to play to your strengths. You should feel comfortable multiplying 2s by 5s to get 10s, so if you extract 2^23*5^23, you can visualize that number: 10^23, which is a 1 followed by 23 zeroes. Then you're left with 5^1*7^3, which you could either multiply out (not fun but not impossible, either) or again repackage to (5)(7) * (7)(7), which is 35 * 49. That is close enough to 35 * 50 that you can quickly see that that number will have 4 digits, so your final number will be those 4 digits followed by 23 zeroes for a total of 27 digits.
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zerosleep
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   03 Feb 2015, 18:00
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Well, 2^23 * 5^24 8 7^3 can be simplified
5 * 7^3 * 10^23

Now either we can multiply 5 and 343 (7^3 = 343) and check or intuitively we can easily see that -
7^3 will be definitely greater than 200 (7*7 = 49 and we have one more 7 to multiply roughly would tak eus to 280+
if you dont remember 7^3 =343)...
what matter here is it will surely give me a -> 3 digit number <-
which when multiplied by 5 will give me no more than a 4 digit number. (we already saw that no is greater than 200 so definitely 4 digit number and not 3)

Hence we can say that on simplifiction we get (4 digit number) * 10^23
This will give me 4 digit number followed by 23 zeroes and hence no of digits will be 27
Ans- :D
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   03 Feb 2015, 22:40
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Hi Bunuel,


We can simplify to: 2^23*5^23*5*7^3= (2*5)^23*5*343=10^23*1715

We can see a pattern in the powers of 10 =>
10^1 has 2 digits
10^2 has 3 digits
....
10^23 has 24 digits

IF we simplify 10^1*1715= 17150 =>1715 adds 3 digits to any number, power of 10 =>

(a number with 24 digits, from 10^23) * ( a number that adds 3 digits) = 27 digits

CORRECT RESPONSE D

Bunuel wrote:
How many digits are there in the product 2^23*5^24*7^3?

A. 24
B. 25
C. 26
D. 27
E. 28


Kudos for a correct solution.
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   24 Jun 2016, 02:58
kdatt1991 wrote:
Bunuel wrote:
How many digits are there in the product 2^23*5^24*7^3?

A. 24
B. 25
C. 26
D. 27
E. 28


Kudos for a correct solution.


Seems like a tricky question, but I hope that I have been able to crack it! Here's my solution:

\((2^{23})*(5^{24})*(7^{3}) = (2^{23})*(5^{23})*(5)*(7)*(7)*(7)\)

\((2^{23})*(5^{23})*(5)*(7)*(7)*(7) = ((2*5)^{23})*(5)*(7)*(7)*(7)\)

\(((2*5)^{23})*(5)*(7)*(7)*(7) = (10^{23})*(35)*(49)\)...... From this step onwards it is probably possible to estimate the number of digits by approximating \((10^{23})*(35)*(49)\) to \((10^{23})*(35)*(50)\)!

But, just to make sure:

\((10^{23})*(35)*(49)\) = \((10^{23})*(35)*(50-1)\).... Therefore \((10^{23})*(1750 - 35)\) which is can be simplified to \((10^{23})*(1715)\)

\((10^{23})*(1715)\) should have exactly 27 digits!

I think the answer is D!

Please consider giving me KUDOS if you felt this post was helpful and correct! or please enlighten me (in case my answer's incorrect) so that I can learn and improve from my mistakes! Thanks. :)


Great Explanation.
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   24 Jun 2016, 04:02
2^23*5^23*5*7^3
or,
10^23*5*343
10^23*1715
=27 digits
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   08 Jul 2017, 09:40
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To find no of digits, first thing comes in our mind is -
either we do multiplication and see for ourselves. But here, looking at the mammoth factors (2^23 * 5^24 8 7^3), we know it is not posssible or at least very time consuming.

SO, next thing comes in our mind is - we know that 10^2 = 100 = 3 digits
10^3 = 1000 = 4 digits
In fact, 10^ n will have n+1 digits.

Now, I see this because I see a lot of 2's and 5's in the option. Hence, let us try to simplify the given equation -
2^23 * 5^24 * 7^3 = (2*5)^23 * 5 * 7^3
= 10^23 * 5 * 7^3 *
= 24 digits + whatever we get from rest

Let us solve the rest,
7^3 will surely give me 3 digit no, if you do not know 7^3 = 343
Multiplying by 5 will give me a 4 digit number.

This will give me 4 digit number followed by 23 zeroes and hence no of digits will be 27.
Ans D
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   19 Aug 2017, 01:36
I did a slightly more lengthy approach (but for someone still struggling with Quant; maybe useful):
- intial amount: 2^23 5^24 7^3
- least number of digits - 2*3*5 = 70 --> 2 digits; leaving me with 70 x 2^22 x 5 23 x 7 ^2
- next number -- 70 x 70 x 2^21 x 5^22 x7
- next number -- 343000 x 2^20 x 5 ^21 (now we can understand from this step only zeroes would get added on)
- we have now 3430000 (7 digits) x 2^19 x 5^20 --> we have now 19 zeroes that get added on; leaving aside 5 to power 1
- so we have 7 + 19 = 25 zeroes and 343*5 = 2 additional digits (1715) that get added on
in total 27 digits
Hope this was helpful to anyone that read this :)
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   26 Jul 2020, 15:49
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Bunuel wrote:
How many digits are there in the product 2^23*5^24*7^3?

A. 24
B. 25
C. 26
D. 27
E. 28


Kudos for a correct solution.

Solution:

Since 2^23 x 5^24 x 7^3 = (2 x 5)^23 x 5 x 7^3 = 10^23 x 1,715, we see that the product is the number 1,715 followed by 23 zeros. Therefore, there are 4 + 23 = 27 digits in the product.

Answer: D
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How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   30 Mar 2023, 04:35
We have \(2^{23}*5^{24}*7^3\)

which can be rewritten as

\(2^{23}*5^{23}*5^1*7^3\)

which gives us

\((2*5)^{23}*5^1*7^3\)

\((10)^{23}*5^1*7^3\)

Now, \(10^{23}\) is going to give us \(1\) followed by \(23\) zeroes, in total \(24\) digits.

Now coming to \(5^1*7^3\).

\(7^3 = 343\) (I happened to know this, but it is better to memorize all cubes up till 10 and squares up till 20 or at least 15 for the GMAT)

Now \(5^1 * 343 = 5*343\) and we can be sure it won't cross 4 digits. It would be approximately \(1500 (5*300) + 200 (5*40) = 1700\), though exactly it will \(1715\).

Now \(1715\) multiplied by \(1\) is going to be \(1715\), thus giving us \(4\) digits in addition to the \(23\) zeroes.

Thus total number of digits is going to be \(4 + 23 = 27\) digits.

Answer is Choice D.
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   02 Apr 2023, 17:28
\(2^{23}*5^{24}*7^3\)

the easiest way is to multiply everything by 2

\(2^{23}*2*5^{24}*7^3\)
\(=2^{24}*5^{24}*7^3\)
\(=(2*5)^{24}*7^3\)
\(=10^{24}*7^3\)

so \(10^{24}\) will have a 1 and 24 ceros, so 25 digits and \(7^3=342\) so 3 digits

because we multiplied by 2 we have to divide by 2 which means one cero less(ex:100/2=50, 1000/2=500...and so on), leaving 24 digits+3 digits=27
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink] New post   05 Apr 2023, 03:55
Maria240895chile wrote:
\(2^{23}*5^{24}*7^3\)

the easiest way is to multiply everything by 2

\(2^{23}*2*5^{24}*7^3\)
\(=2^{24}*5^{24}*7^3\)
\(=(2*5)^{24}*7^3\)
\(=10^{24}*7^3\)

so \(10^{24}\) will have a 1 and 24 ceros, so 25 digits and \(7^3=342\) so 3 digits

because we multiplied by 2 we have to divide by 2 which means one cero less(ex:100/2=50, 1000/2=500...and so on), leaving 24 digits+3 digits=27


Very clever! Also , one small mistake \(7^3 = 343\) and not \(342\).
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]
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