lqc172 wrote:I still don't quite get this question...
Does "least possible" integer here means an integer that is possible but the smallest?
Lets breakdown the question:
n is a positive integer
Therefore n>0
the product of all the integers from 1 to n, inclusive, is a multiple of 990
Therefore (1 x 2 x 3 ..... x n) / 990 gives an integer
Therefore n!/990 gives an integer
Therefore n!/(9 x 10 x 11)gives an integer
Question: What is the least possible value of n!
This means what is the lowest value that n! can be so that when it is divided by 9 x 10 x 11, the outcome gives an integer
The lowest value of n! should have all the factors that are in the demoninator so that when divided an integer is formed
11! / (9 x 10 x 11) = (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (9 x 10 x 11)
= 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
10! / (9 x 10 x 11) = (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (9 x 10 x 11)
= (9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / 11 = mixed number
Any number lower than 10! will give a mixed number.
Any number higher that 11! will give an integer, however 11! is the lowest value that will produce an integer when 990 is the denominator.
