multiples

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divya23: Master | Next Rank: 500 Posts; Posts: 113; Joined: Mon Apr 26, 2010 8:58 amFollowed by: 14 members; Thanked: 2 times

by divya23 » Thu Jun 09, 2011 7:36 am

if n is a +ve int and product of all d int from 1 to n inclusive is a multiple of 990 find least value of n

[spoiler]ans = 11[/spoiler]

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cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 amFollowed by: 123 members; Location: India; Thanked: 310 times; GMAT Score:750

by cans » Thu Jun 09, 2011 7:41 am

n! is multiple of 990
990=11*10*9
11 is a prime number, and thus n! must contain 11 so that it is multiple of 990
thus least value of n=11

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manpsingh87: Master | Next Rank: 500 Posts; Posts: 436; Joined: Tue Feb 08, 2011 3:07 amFollowed by: 6 members; Thanked: 72 times

by manpsingh87 » Thu Jun 09, 2011 7:43 am

divya23 wrote:if n is a +ve int and product of all d int from 1 to n inclusive is a multiple of 990 find least value of n

[spoiler]ans = 11[/spoiler]

product of all integers from 1 to n=n!;
990=11*3^2*2*5;
now largest prime no. in 990 is 11, also since 990, is a factor of n! therefore minimum value of n is 11

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by [email protected] » Fri Jun 10, 2011 10:18 pm

Guyzzz... please tell me why can't 2 or 3 be the answer for this question.

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by Anurag@Gurome » Fri Jun 10, 2011 10:37 pm

[email protected] wrote:Guyzzz... please tell me why can't 2 or 3 be the answer for this question.

Let's say n = 3
Then, the product of all the integers from 1 to n inclusive = 1*2*3 = 6
Now 6 is not a multiple of 990.

The product of all the integers from 1 to n inclusive, i.e. n! will be a multiple of 990 if and only if all the factors of 990 is contained within n!

Now, 990 = 9*10*11 = 2*(3^2)*5*11
Hence, largest prime factor of 990 is 11.
Therefore, n! must contain 11 within itself.
Now, if n = 11, n! = 11*10*9*8*...*2*1 = 990*(Something)

Hence, least possible value of n is 11.

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