China | A Nice Factorial Problem

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[@kiyotaka-z6q]

Me after seeing this video I think I am a master of algebra😂

[@kiyotaka-z6q]

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[@echandler]

3024 = 2^4 * 3^3 * 7 = product of 4 consecutive integers. 5 is not a factor, thus 5 and 10 are excluded. Hence 6*7*8*9. Thus x = 2.

[@adamnyback]

3024^(1/4)≈7.4
The four consecutive whole numbers around 7.4 are 6, 7, 8 and 9.

[@gilbertbarnett8225]

I wish I had this type of substitution problem when I was in high school. My teacher would criticize that I didn't use substitution as a method and therefore wasn't "showing my work" because I felt that y=2x wasn't worthy of making a new variable if I could keep track of it without.

[@Garden_Of_Eden_]

You are a senior mathematician

[@juergenilse3259]

(x+7)!/(x+3)!=(x+7)*(x+6)*(x+5)*(x+4) according to the definition of factorial.
So we seach for 4 consecutive natural numbers which product is 3024. 3024 is not dividable by 5, so none of the 4 numbers can be dividable by 5. All numbers must be smaller than 10, because the lowest sequence of 4 consecutive numbers, from which none is dividable by 5 and at lleast one of them is greater or equal to 10 is 11, 12, 13, 14, but the product of those 4 numbers is greater than 10^4=10000, so the product would be much greater than 3024, if at lleast one number is greater than 10. There are 2 sequences of 4 consecutive natural numbers smaller than 10, for which none of the numbers is dividablle by 5:: 1, 2, 3, 4 and 6, 7, 8, 9.
1*2*3*4 can not be of the form (x+7)!/(x+3)!, because than, we would have (--3)! in the denoinator, which is undefined So the only solution (if thhere is a solution) can be (x+7)=9, which means x=2.
After llooking the video, i would say, that my solution is much simpler ....
But even if we try to find the soutionike in the video.: why not set t=x^2+11x+29? Then we get
(t-1)*(t+1)=3024
t^2--1=3024
t^2=3025
Since 3025 is dividabe by 5 (last digit is 5) and b 11 (alternate cross su is 0), we now, that the square root (if it is a whoe number) must be dividabe by 5*11=55. /2*55)^2 is mmuch greater than 3025, so t must be 55 (since t ust be a whoe number).
t=55 or t=-55 (rejected since x^2+11x+29 is positive for each naturall number x)
Revert substitution::
x^2+11x+29=55
x^2+11x--26=0
We come easier to this equation with substituting t=x^2+11x+29 rather than t=x^2+11x+28.