SOLUTION: I need explanation and solution: The product of four consecutive natural numbers is 255024. Find these numbers.

Question 861357: I need explanation and solution:
The product of four consecutive natural numbers is 255024. Find these numbers.

Found 2 solutions by mananth, richwmiller:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
let the natural numbers be x,x+1,x+2,x+3
The product is 255024

x(x+1)(x+2)(x+3)=255024
rearrange
x(x+3)(x+1)(x+2)=255024
(x^2+3x)(x^2+3x+2)=255024
let x^2+3x=a
a(a+2)=255204
a^2+2a= 255024
a^2+2a+1=255024+1
(a+1)^2= 255025
take the square root
(a+1) = +/- 505
a=504 OR -506
substitute a
x^2+3x-504 =0
x^2+24x-21x-504=0
x(x+24)-21(x+24)=0
(x+24)(x-21)=0
x= 21 a natural number
OR
x^2+3x=-506
x^2+3x+506=0
the roots are not real
Hence x=21
21,22,23,24 are the numbers

Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
4th root of 255024 is about 22.47
so 22 or 23 or both are factors
255024/23=11088
11088/22=504
so both 22 and 23 are factors
since we want 4 consecutive numbers the factors must be
20,21,22,23,
21,22,23,24
22,23,24,25
21 or 24 or both are factors
let's test 21
504/21=24
the factors are 21,22,23,24

RELATED QUESTIONS
I need explanation and solution: The product of four consecutive natural numbers is... (answered by rothauserc)
I need explanation: The product of four consecutive natural numbers is 255024. Find... (answered by Alan3354)
The product of four consecutive natural numbers is 255024. Please, help to find these... (answered by Alan3354)
The product of two consecutive natural numbers is 1056. Find the sum of these two... (answered by Earlsdon)
The product of four consecutive natural numbers is 1680 find the... (answered by josgarithmetic,ikleyn)
the product of four consecutive natural numbers is 5040.find those... (answered by Alan3354)
the product of four consecutive natural numbers which are multiple of 5 is 15000 . find... (answered by Alan3354)
The product of four consecutive natural numbers which are multiple of five is 15000 .... (answered by Alan3354)
the product of two consecutive even natural numbers is 48. find the numbers. (answered by richard1234,Alan3354)