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Find the smallest value of n

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,683
Find the smallest value of $n$ for which $\dfrac{n!\cdot n!}{(n+6)!}$ is an integer.
 

kaliprasad

Well-known member
Mar 31, 2013
1,309
Find the smallest value of $n$ for which $\dfrac{n!\cdot n!}{(n+6)!}$ is an integer.
89

we need (n+1) to (n+6) all should be composite so product should be a factor of n!. then check that it is factor smallest 6 composite consecutive numbers start at 90

90 = 9 * 10
91 = 13 * 7
92 = 23 * 4
93 = 3 * 31
94 = 2 * 47
95 = 5 * 19

take the product and we have 10 different numbers < 90 on the right and product divided 89!
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,683
89

we need (n+1) to (n+6) all should be composite so product should be a factor of n!. then check that it is factor smallest 6 composite consecutive numbers start at 90

90 = 9 * 10
91 = 13 * 7
92 = 23 * 4
93 = 3 * 31
94 = 2 * 47
95 = 5 * 19

take the product and we have 10 different numbers < 90 on the right and product divided 89!
Well done, kaliprasad!:cool: