K(N) denotes the number of ways in which N can be expressed as a difference of two perfect squares. which of the following is maximum?

1) K(110)
2) K(105)
3) K(216)
4) K(384)

• Suppose n is a product, PQ, where P≥Q.
  Then we want integers a and b so that: .a^2−b^2=PQ
  so (a+b)(a-b)=PQ
  We assume that a+b=P
  a-b=Q
  And by solving we get a=(P+Q)/2
  b=(P-Q)/2
  since a and b are integers, P and Q must be Parity.
  Both are Even or Both are Odd
  Now,
  n=110=2⋅5⋅11
  110 cannot be factored into two factors with the same parity.
  Hence, 110 cannot be expressed as a difference of squares.
  K(110)=0
  
  n=105=3⋅5⋅7
  105 has four possible factorings.
  P Q a b
  105 1 53 52
  35 3 19 16
  21 5 13 8
  7 15 11 4
  
  n=216=2^3⋅3^3
  216 has four possible factorings.
  P Q a b
  108 2 55 53
  54 4 29 25
  36 6 21 15
  18 12 15 3
  
  n=384=2^7⋅3
  384 has six possible factorings.
  P Q a b
  192 2 97 95
  96 4 50 46
  64 6 35 29
  48 8 28 20
  32 12 22 10
  24 16 20 4
  Therefore K(384) has Max number of possibilities
• 6 years agoHelpfull: Yes(0) No(0)