240 = 2^4 * 3 * 5

That could be 2 * 2 * 2 * 2 * 3 * 5 which could imply the number sought was

2^4 * 3^2 * 5 * 7 * 11 * 13 = 720,720

(Each factor of 2 in 240 implies a prime factor to the 1st power; a factor of 3 in 240 implies a prime factor squared; the factor of 5 in 240 implies a prime factor to the 4th power.)

There could be alternate factorings of 240, such as 2 * 2 * 3 * 4 * 5, which would correspond to the number's being

2^4 * 3^3 * 5^2 * 7 * 11 = 831,600

Clearly the  720,720 is the lower of the two and in fact this program verifies it's the lowest of such reworkings:

DefDbl A-Z

Dim crlf$, fct(20, 1)

Private Sub Form_Load()

 Form1.Visible = True

 

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 For n = 1 To 1000000

   f = factor(n)

   divisors = 1

   For i = 1 To f

     divisors = divisors * (fct(i, 1) + 1)

   Next

   If divisors = 240 Then

     Text1.Text = Text1.Text & n & crlf

       For i = 1 To f

         Text1.Text = Text1.Text & "     " & fct(i, 0) & Str(fct(i, 1)) & crlf

       Next

   End If

   DoEvents

 Next

 

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

Function factor(num)

 diffCt = 0: good = 1

 n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0

 If limit <> Int(limit) Then limit = Int(limit + 1)

 dv = 2: GoSub DivideIt

 dv = 3: GoSub DivideIt

 dv = 5: GoSub DivideIt

 dv = 7

 Do Until dv > limit

   GoSub DivideIt: dv = dv + 4 '11

   GoSub DivideIt: dv = dv + 2 '13

   GoSub DivideIt: dv = dv + 4 '17

   GoSub DivideIt: dv = dv + 2 '19

   GoSub DivideIt: dv = dv + 4 '23

   GoSub DivideIt: dv = dv + 6 '29

   GoSub DivideIt: dv = dv + 2 '31

   GoSub DivideIt: dv = dv + 6 '37

   If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function

 Loop

 If n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1

 factor = diffCt

 Exit Function

DivideIt:

 cnt = 0

 Do

  q = Int(n / dv)

  If q * dv = n And n > 0 Then

    n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) Else limit = 0

    If limit <> Int(limit) Then limit = Int(limit + 1)

   Else

    Exit Do

  End If

 Loop

 If cnt > 0 Then

   diffCt = diffCt + 1

   fct(diffCt, 0) = dv

   fct(diffCt, 1) = cnt

 End If

 Return

End Function

The output being the following, where the valid number with 240 divisors is on the first line of each group followed by a list of its prime factors, with the power to which that factor is raised.

Second place is indeed the 

831,600 = 2^4 * 3^3 * 5^2 * 7 * 11

we found above, resulting from a factoring of 240 = 5 * 4 * 3 * 2 * 2

720720

     2 4

     3 2

     5 1

     7 1

     11 1

     13 1

831600

     2 4

     3 3

     5 2

     7 1

     11 1

942480

     2 4

     3 2

     5 1

     7 1

     11 1

     17 1

982800

     2 4

     3 3

     5 2

     7 1

     13 1

997920

     2 5

     3 4

     5 1

     7 1

     11 1