Product of prime numbers

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Deepthi Subbu: Master | Next Rank: 500 Posts; Posts: 298; Joined: Tue Feb 16, 2010 1:09 amFollowed by: 1 member; Thanked: 2 times

by Deepthi Subbu » Tue Mar 12, 2013 8:50 am

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

In OG13, the answer is calculated based on multiplication. Is there a short cut?

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Source: Beat The GMAT — Problem Solving | Tue Mar 12, 2013 8:50 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Mar 12, 2013 8:55 am

Deepthi Subbu wrote:The product of all prime numbers less than 20 is closest to which of following powers of 10 ?
(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

I added the answer choices.
Since the numbers are very spread apart (each answer choice is 10 times greater than the next answer choice), we can be somewhat aggressive with our estimation.

We have the product (2)(3)(5)(7)(11)(13)(17)(19)

Let's see if we can group the numbers to get some approximate powers of 10

First (2)(5)=10, so we get (2)(3)(5)(7)(11)(13)(17)(19) = (10)(3)(7)(11)(13)(17)(19)

Next, 11 is close enough to 10, so we get: (10)(3)(7)(11)(13)(17)(19) = (10)(3)(7)(10)(13)(17)(19) [approximately]

Next, (7)(13)=91, which is pretty close to 100. So we get (10)(3)(7)(10)(13)(17)(19) = (10)(3)(100)(10)(17)(19) [approximately]

Finally, 3(17)=51, and (51)(19) is very close to (51)(20), which is very close to 1000
So,(10)(3)(100)(10)(17)(19) = (10)(1000)(100)(10)= 10,000,000 [approximately]

Since 10,000,000 = 10^7, the best answer is C

Cheers,

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Mar 12, 2013 1:18 pm

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

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

Since the answer choices are VERY far apart, we can BALLPARK.
For every value that we round UP, we should compensate by rounding another value DOWN.

2*3*5*7*11*13*17*19
210 * 10*15 * 15*20
200*150*300 = 9,000,000.

The closest power of 10 = 10,000,000 = 10^7.

The correct answer is C.

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by [email protected] » Thu Feb 15, 2018 4:23 pm

Hi All,

We're asked to determine which of the 5 answers is CLOSEST to the product of all the prime numbers less than 20. We can use a mix of Arithmetic and estimation to answer this question.

To start, it's worth noting that the Answers are NOT "close together" - each answer is 10 TIMES greater than the next, so we really just have to get 'close enough' with our math to answer this question.

The primes that are less than 20 are: 2, 3, 5, 7, 11, 13, 17 and 19

(5)(19) = 95 --> about 100 = 10^2
(2)(3)(17) = 102 --> about 100 = 10^2
(7)(13) = 91 --> about 100 = 10^2
(11) = about 10 = 10^1

(10^2)(10^2)(10^2)(10^1) = 10^7

Final Answer: C

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Rich

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by Scott@TargetTestPrep » Tue Feb 20, 2018 4:25 pm

Deepthi Subbu wrote:The product of all prime numbers less than 20 is closest to which of following powers of 10 ?

2 x 3 x 5 x 7 x 11 x 13 x 17 x 19

Let's group some of these numbers to get powers of 10:

2 x 5 = 10 = 10^1

7 x 11 x 13 = 1001 â‰ˆ 10^3

Now we are left with 3, 17 and 19:

3 x 17 x 19 = 51 x 19 â‰ˆ 50 x 20 = 1000 = 10^3

Thus the product of all the prime numbers less than 20 is closest to 10^1 x 10^3 x 10^3 = 10^7.

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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