The product of four consecutive natural numbers is 360. find the numbers.

Answer :

We know natural numbers  " The whole numbers from 1 upwards: 1, 2, 3, ...  and no negative numbers and no fractions. "

Let Four consecutive natural numbers  =  x  , ( x +  1 )   , (  x  +  2 ) , ( x + 3 )  , Here x  , can't be negative  as we know natural numbers doesn't have negative numbers .

From question we get

  x  ( x +  1 ) ( x  +  2 ) ( x + 3 ) =  360

$\Rightarrow$x  ( x + 3 ) ( x +  1 ) ( x  +  2 ) =  360

$\Rightarrow$ ( x² +  3x ) ( x² +  3x +  2 ) =  360

$\Rightarrow$ [ ( x² +  3x +  1 ) -  1 ] [ ( x² +  3x +  1 ) +  1 ]  =  360

$\Rightarrow$ [ ( x² +  3x +  1 )² -  1² ]  =  360

$\Rightarrow$ ( x² +  3x +  1 )² -  1  =  360

$\Rightarrow$ ( x² +  3x +  1 )²  =  361

$\Rightarrow$ ( x² +  3x +  1 )  =  19

$\Rightarrow$ x² +  3x  - 18 =  0

$\Rightarrow$ x² +  6x - 3x- 18 =  0

$\Rightarrow$ x ( x +   6 ) - 3 ( x + 6 )  =  0

$\Rightarrow$ ( x - 3 ) ( x +   6 ) =  0
So,
x  =  3 and  -  6  , We neglect x =  - 6 ,  As we assumed x as a natural number .
So,
x =  3 ,
x  +  1  =  3 +  1 = 4 ,
x  +  2  =  3 +  2 = 5
And
x  +  3  =  3 +  3 = 6 ,

And
Four consecutive natural numbers whose product is 360 = 3 , 4 , 5 and  6                                                                              ( Ans )