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# The product of all the prime numbers less than 20 is closest

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The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 02:21
Expert's post
24
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Difficulty:

85% (hard)

Question Stats:

48% (02:16) correct 52% (01:16) wrong based on 1016 sessions

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The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

Diagnostic Test
Question: 15
Page: 22
Difficulty: 650
[Reveal] Spoiler: OA

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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 02:22
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SOLUTION

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

We should find the approximate value of 2*3*5*7*11*13*17*19 to some power of 10.

# of different approximations are possible.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

Approach #2:

2*5=10
3*17=~50 (actually more than 50);
7*13=~100 (actually less than 100);
11*19=~200 (actually more than 200)

$$2*3*5*7*11*13*17*19\approx{10*50*100*200}=10^7$$.

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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 03:17
Hi,

Difficulty level: 650

Product of prime numbers less than 20 = 2*3*5*7*11*13*17*19
=(2*5)*3*(7*13)*(11*17)*19
=3*10*91*189*19
~(30)*(90*19)*190
~5700*1710
~5700*1700
~9690000 $$(=10^6)$$

Regards,

Last edited by cyberjadugar on 02 Jul 2012, 03:39, edited 2 times in total.
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 03:25
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Bunuel wrote:
The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

Diagnostic Test
Question: 15
Page: 22
Difficulty: 650

GMAT Club is introducing a new project: The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide for GMAT® Review, 13th Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

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Thank you!

The product is $$2*3*5*7*11*13*17*19 = 10 * 21 * 11*(13*17) * 19 \approx10*20*10*15^2*20=900*10^4\approx10^7$$.

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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 03:45
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Hi,

The magnitude of error depends on the approximation done in the question.

For example: 51*671=34221
50*671=33550 (error = 671)
51*670=34170 (error = 51)

Thus, the when the larger number is approximated the error is less.

Regards,
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 05:53
Hi,

The magnitude of error depends on the approximation done in the question.

For example: 51*671=34221
50*671=33550 (error = 671)
51*670=34170 (error = 51)

Thus, the when the larger number is approximated the error is less.

Regards,

then why D instead of C.
i got the same and chose D
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 05:58
kashishh wrote:
Hi,

The magnitude of error depends on the approximation done in the question.

For example: 51*671=34221
50*671=33550 (error = 671)
51*670=34170 (error = 51)

Thus, the when the larger number is approximated the error is less.

Regards,

then why D instead of C.
i got the same and chose D

Hi,

I believe you are right and answer should be (C).

Regards,
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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02 Jul 2012, 12:58
2*3*5*7*11*13*17*19 = 210*(10+1)(10+3)(10+7)(10+9)
210 ~ 10*10*2
So the original equation will be equivalent to 10*10*2*(10+1)(10+3)(10+7)(10+9)
The highest order of the equation will be 10^6, and other part of expression will not make it closer to 10^7.
I have solved it mostly on hunch, no sure how correct I am.
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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06 Jul 2012, 03:12
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Expert's post
SOLUTION

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

We should find the approximate value of 2*3*5*7*11*13*17*19 to some power of 10.

# of different approximations are possible.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

Approach #2:

2*5=10
3*17=~50 (actually more than 50);
7*13=~100 (actually less than 100);
11*19=~200 (actually more than 200)

$$2*3*5*7*11*13*17*19\approx{10*50*100*200}=10^7$$.

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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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25 Oct 2013, 04:45
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I have approached in the following way:-

2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 = 10 x 21x 11 x 13 x 17 x 19 = 10 x 10 x 2.1 x 10 x 1.1 x 10 x 1.3 x 10 x 1.7 x 10 x 1.9 = 10^6 x (2.1 x1.1 x 1.3 x 1.7 x 1.9)

The approximation of the (2.1 x 1.7 x 1.9 x 1.3 x 1.1) will be close to 10, so the answer is 10^7
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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05 Mar 2014, 11:48
Similar to Bunuel's first approach:
I thought it was easiest to just do :
2*3*5*7 = 210
11 roughly 10
13 roughly 10
17 roughly 20
19 roughly 20
so 210*10*10*20*20 = 8*10^6 = 10^7
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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06 Mar 2014, 21:40
Product of Prime no between 1 & 20 =

(2*3*5*7)*(11*13)* (17*19)

210 * 143 * 323

$$(10^2+10) * (10^1+43) * (10^3+23)$$

$$(10^3+4300+10^2+430) * (10^3+23)$$

$$5830 * (10^3+23)$$

5830000 + 134090

= 6974090 this is near to 10^7

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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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10 Mar 2014, 10:15
2*19 =~40
3*17=~60
5*13=~50 (More than 50)
7*11=~70 (")

=40*60*50*70
=4*6(10^2)*5*7(10^2)
=2*(10^1*10^2)*3.5(10^1*10^2)
=7*10^6
=~10^7

Like this any prime number can be multiplied with.
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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11 Apr 2014, 19:07
Can anyone comment on my method's effectiveness/efficiency:
Prime numbers < 20 are: 2, 3, 5, 7, 11, 13, 17, 19 (Count = 8)
Median = 9 (through (7+11)/2)
So product should be approx. close to 9^8 which in turn is closest to 10^7, thus answer is 10^7.

Thanks for reviewing.
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The product of all the prime numbers less than 20 is closest [#permalink]

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21 Jun 2015, 03:36
Here's my solution
2*3*5*7*11*13*17*19

1) 2*3*5*19 ~ 30*20
2) 11*13 ~ 140
3) 7*17 ~ 120

SO we have 600*140*120 = 10^4*168*6 (168*6 ~ 1000) --> ~ 10^4*10^3 ~10^7

Method 2

2*3*5*7*11*13*17*19 = 2*5*11*19*(10-7)(10+7)(10-3)(10+3) ~ 2*10*10*10*50*80=100*10*10*10*10*8 ~$$10^7$$ Answer (D)
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Re: The product of all the prime numbers less than 20 is closest [#permalink]

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01 Aug 2016, 20:48
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Re: The product of all the prime numbers less than 20 is closest   [#permalink] 01 Aug 2016, 20:48
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