If n is a positive integer, and the product of all integers 1 to n, inclusive, is a multiple of 990, what is?

If n is a positive integer, and the product of all integers 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, how am I supposed to arrive at that answer?

1 Answer

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  • 9 years ago
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    To be a multiple of 990, the product must contain sufficient prime factors of 990. Let's factor 990:

    990 = 9*10*11 ...

    ... wait a minute! We can stop right there. We have to get at least as high as 11 to have that prime factor of 11 in the product. Since this product will also include factors of 9 and 10, it must be a multiple of 990. Specifically,

    1*2*3*4*5*6*7*8*9*10*11 =

    1*2*3*4*5*6*7*8* 990,

    and no smaller factorial will do

    (N factorial, written N!, is the product of the numbers 1 through N)

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