1 ! × 2 ! × ⋯ × n ! = 2 8 8
What is the value of n satisfying the equation above?
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Thank you for your solution!
We can also do this question by prime factorizing 288, that is, taking the factors of 2 and three, when we do this we get 2 5 and 3 2 as it's factors, this is only possible when n = 4 , as when n = 4 the power of 2 is 5 and that of 3 is 2
Just do trial and error.
Divided 2 8 8 by 1 ! then by 2 ! then by 3 ! and the result is 2 4 which is 4 ! .
Any generalization possible here ?
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5 ! = 1 2 0 , which does not divide 288.
2 ! = 2 which does divide 288, but 1 ! × 2 ! = 2 = 2 8 8
3 ! = 6 which divides 288, but 1 ! × 2 ! × 3 ! = 1 2 = 2 8 8
4 ! = 2 4 which divides 288. 1 ! × 2 ! × 3 ! × 4 ! = 2 8 8 .
Answer = 4