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

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Math Expert
Joined: 02 Sep 2009
Posts: 41912

Kudos [?]: 129370 [0], given: 12197

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

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03 Feb 2015, 09:53
Expert's post
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Difficulty:

45% (medium)

Question Stats:

60% (00:55) correct 40% (01:14) wrong based on 175 sessions

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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

Kudos for a correct solution.
[Reveal] Spoiler: OA

_________________

Kudos [?]: 129370 [0], given: 12197

Intern
Joined: 19 Sep 2014
Posts: 22

Kudos [?]: 64 [8], given: 7

Concentration: Finance, Economics
GMAT Date: 05-05-2015
Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]

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03 Feb 2015, 11:41
8
KUDOS
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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.

Kudos [?]: 64 [8], given: 7

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

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03 Feb 2015, 18:00
3
KUDOS
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]

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03 Feb 2015, 22:40
2
KUDOS
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.

_________________

THANKS = KUDOS. Kudos if my post helped you!

Napoleon Hill — 'Whatever the mind can conceive and believe, it can achieve.'

Kudos [?]: 28 [2], given: 73

Math Expert
Joined: 02 Sep 2009
Posts: 41912

Kudos [?]: 129370 [1], given: 12197

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

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09 Feb 2015, 04:57
1
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Expert's post
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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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Re: How many digits are there in the product 2^23*5^24*7^3? [#permalink]

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24 Jun 2016, 02:57
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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, 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.
+1 Kudos
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Manickam

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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, 04:02
2^23*5^23*5*7^3
or,
10^23*5*343
10^23*1715
=27 digits

Kudos [?]: 100 [0], given: 10

Intern
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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, 09:40
3
KUDOS
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]

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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

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Re: How many digits are there in the product 2^23*5^24*7^3?   [#permalink] 19 Aug 2017, 01:36
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