Optimizing n for Integer Solutions in Factorial Expressions

[anemone]

Find the smallest value of $n$ for which $\dfrac{n!\cdot n!}{(n+6)!}$ is an integer.

 

[kaliprasad]

Find the smallest value of $n$ for which $\dfrac{n!\cdot n!}{(n+6)!}$ is an integer.

89

Spoiler

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!

 

[anemone]

89

Spoiler

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!