Lowest common multiple question

Question

Find the positive integer n such that the least common multiple of n and n-30 is n+1320.

Answer 1

Hi Ethan,

 

Normally, when I look at least common multiples I first factor the items.  In this case, we have n and n-30.  There is really nothing to factor.

 

So, let's multiply the two factors and set it equal to the given LCM.

 

n(n-30) = n+1320

 

n² - 30n = n+1320 and solve for n

 

n² - 31n - 1320 = 0

 

Use the quadratic formula to solve for n

 

+31 ±√[31²-(4)(1)(-1320)]

--------------------------------

            (2)(1)

=31 ±√[961 + 5280]

  -----------------------

             2

= 31 ± 79

  ----------

        2

 

= 110/2 , -48/2

= 55, -24

 

But we need a positive integer so n = 55

 

We can check this

 

55(55-30) = 55+1320