Is There A Factorials Table?

1 ! × 2 ! × × n ! = 288 \large 1!\times2! \times\cdots\times n! = 288

What is the value of n n satisfying the equation above?

2 3 4 5

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4 solutions

Mehul Arora
Mar 18, 2016

5 ! = 120 5! = 120 , which does not divide 288.

2 ! = 2 2! =2 which does divide 288, but 1 ! × 2 ! = 2 288 1! \times 2! =2 \neq 288

3 ! = 6 3! = 6 which divides 288, but 1 ! × 2 ! × 3 ! = 12 288 1! \times 2! \times 3! = 12 \neq 288

4 ! = 24 4! =24 which divides 288. 1 ! × 2 ! × 3 ! × 4 ! = 288. 1! \times 2! \times 3! \times 4! = 288.

Answer = 4 \boxed {4}

Thank you for your solution!

Chung Kevin - 5 years, 2 months ago

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 2^5 and 3 2 3^2 as it's factors, this is only possible when n = 4 n=4 , as when n = 4 n=4 the power of 2 is 5 and that of 3 is 2

Racchit Jain - 4 years, 11 months ago

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Who'll waste all that time? :P

Mehul Arora - 4 years, 11 months ago

Just do trial and error.

Adrian Florin
Mar 25, 2016

Divided 288 288 by 1 ! 1! then by 2 ! 2! then by 3 ! 3! and the result is 24 24 which is 4 ! 4! .

Suneel Kumar
Mar 25, 2016

Any generalization possible here ?

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