Primes: The product of all the prime numbers less than 20 is

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II: Master | Next Rank: 500 Posts; Posts: 400; Joined: Mon Dec 10, 2007 1:35 pm; Location: London, UK; Thanked: 19 times; GMAT Score:680

by II » Mon Mar 17, 2008 6:31 am

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

A) 9
B) 8
C) 7
D) 6
E) 5

Instead of just listing out the prime numbers under 20, and then multiplying them together, is there another quicker way to solve this.

2x3x5x7x11x13x17x19 can take up a huge portion of the 2 minutes for the question.
Is there another quicker way to solve this ?

Thanks.
II

Last edited by II on Tue Apr 29, 2008 5:28 am, edited 1 time in total.

Source: Beat The GMAT — Problem Solving | Mon Mar 17, 2008 6:31 am

mnjoosub: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Wed Feb 27, 2008 9:32 pm; Location: Mauritius; Thanked: 2 times

by mnjoosub » Mon Mar 17, 2008 6:45 am

Just reduce all the prime numbers to the power of 10, then add the power only, we don't mind with the exact result.

i.e. 2*3*5*7*11*17*19 = 10*21*11*17*19

this is eaual to
1..0*2.1*1.1*1.7*1.9 * 10^6

= 9....*10^6

Ans = D

II: Master | Next Rank: 500 Posts; Posts: 400; Joined: Mon Dec 10, 2007 1:35 pm; Location: London, UK; Thanked: 19 times; GMAT Score:680

by II » Mon Mar 17, 2008 10:15 am

Thanks for responding ... but your answer is incorrect. The OA is C.

2*3*5*7*11*13*17*19 = 9,699,690. This is closest to 10,000,000 which is 10^7. So the answer is c.

But I want to see if there was a shortcut or quicker way to solve this other than just doing mental multiplication of all those numbers ... providing a quicker and more efficient way to answer this.

xilef: Master | Next Rank: 500 Posts; Posts: 111; Joined: Thu Jan 31, 2008 4:05 pmFollowed by: 1 member; Thanked: 18 times

by xilef » Mon Mar 17, 2008 2:49 pm

mnjoosub had the right idea

Not sure if there is a quicker way besides approximation:

2*3*5*7*11*13*17*19 = 10*21*11*13*17*19 = 10*(10*2.1)*(10*1.1)*(10*1.3)*(10*1.7)*(10*1.9) = 10*10*10*10*10*10*9.7 ~ 10^7

II: Master | Next Rank: 500 Posts; Posts: 400; Joined: Mon Dec 10, 2007 1:35 pm; Location: London, UK; Thanked: 19 times; GMAT Score:680

by II » Wed Mar 19, 2008 8:07 am

Hi there ... I see where you are going with it ... but you still need to multiply 2.1, 1.1, 1.3, 1.7, 1.9 ... to get the 9.7.

I suppose you know that at the least it will be 10^6. And can guesstimate the multiplication of the above numbers to be close to 10 (9.7) ... which enables us to reach the answer 10^7.

Thanks.

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