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\(((2*5)^{23})*(5)*(7)*(7)*(7) = (10^{23})*(35)*(49)\)...... 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.

Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]

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03 Feb 2015, 17:00

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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

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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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]

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24 Jun 2016, 01:57

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\(((2*5)^{23})*(5)*(7)*(7)*(7) = (10^{23})*(35)*(49)\)...... 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. +1 Kudos
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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]

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08 Jul 2017, 08:40

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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

Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]

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19 Aug 2017, 00:36

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 :)