Multiples and Factors

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Multiples and Factors

by Aishwarya1204 » Tue Oct 16, 2012 11:36 pm
If n is a positive integer and the product of all the integers from 1 to n inclusive is a multiple of 990, what is the least possible value of n.

a) 10
b) 11
c) 12
d) 13
e) 14

The answer is b

I came to this answer simply by noticing that 11 is a factor of 990 and its a prime so we would definitely need 11 to be in the numbers ( 1 to n ).

Is there an actual method to this question?
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by Anurag@Gurome » Tue Oct 16, 2012 11:47 pm
Aishwarya1204 wrote:If n is a positive integer and the product of all the integers from 1 to n inclusive is a multiple of 990, what is the least possible value of n.

a) 10
b) 11
c) 12
d) 13
e) 14

The answer is b

I came to this answer simply by noticing that 11 is a factor of 990 and its a prime so we would definitely need 11 to be in the numbers ( 1 to n ).

Is there an actual method to this question?
It is given that n! = 990 * a, for some integer a.
Then n! = 2 * 3² * 5 * 11 * a, which implies that n! should have all factors of 990 so that all factors are multiples of 990.
So, it should have 11 also, which means the least value of n is 11.

The correct answer is B.
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