for which of the following n is the number 2^74 + 2^2058 + 2^2n is a perfect square

• 2^74 + 2^2058 + 2^2n =(2^37)^2 + (2^1029)^2 + (2^n)^2
  now, if we put 2^37 = a ; 2^1029 = b; Then for the above expression to be perfect square 2^2n must be equal to (2*a*b)= 2*(2^37)*(2^1029);
  ==> 2^2n = 2^(1067)
  ==> 2n = 1067 ,
  but this case is not possible since R.H.S is an odd integer whereas L.H.S is an even integer.
  So , the above mentioned case can't hold.
  
  Now,if we put 2^37 = a; 2^n = b ; So, for the given expression to be perfect square 2^2058 = (2*a*b)= 2*(2^37)*(2^n) = 2^(n+38);
  So, 2058 = (n+38)
  => n = 2020
  So,The answer is n = 2020
• 11 years agoHelpfull: Yes(30) No(6)
• We know that a^2 + 2ab + b^2 = (a+b)^2
  Here, 2^74 + 2^2058 + 2^2n = (2^37)^2 + 2*2^37*2^2020 + (2^n)^2
  If the number is a perfect square, then 2^n should equal 2^2020.
  Hence, n=2020
• 11 years agoHelpfull: Yes(21) No(8)
• Given expression 2^74 + 2^2058 + 2^2n
  
  We can write the above expression in the form of (a+b)^2
  
  2^74 + 2^2058 + 2^2n
  
  =(2^37)^2 + 2 X 2^37 X 2^n + (2^n)^2
  
  =(2^37)^2 + 2^(38 + n) + (2^n)^2
  
  So , we can write
  
  (38 + n) = 2058
  
  => n = 2020
• 5 years agoHelpfull: Yes(2) No(1)
• 2^74 +2^2058+2^2n = K^2
  2^74 +2^2058+2^2n = (2^37)^2+2^2058+(2^n)^2
  We try to write this expression as (a+b)^2=a^2+2ab+b^2
  Now a = 2^37, 2ab = 2^2058 and b = 2^n
  Substituting the value of a in 2ab, we get b = 2020
• 8 years agoHelpfull: Yes(0) No(3)
• 2^74 +2^2058+2^2n =
  K
  2
  K2
  2^74 +2^2058+2^2n =
  (
  2
  37
  )
  2
  +
  2
  2058
  +
  (
  2
  n
  )
  2
  (237)2+22058+(2n)2
  We try to write this expression as
  (
  a
  +
  b
  )
  2
  =
  a
  2
  +
  2
  a
  b
  +
  b
  2
  (a+b)2=a2+2ab+b2
  Now a =
  2
  37
  237, 2ab =
  2
  2058
  22058 and b =
  2
  n
  2n
  Substituting the value of a in 2ab, we get b = 2020
• 6 years agoHelpfull: Yes(0) No(2)
• 2^74 +2^2058+2^2n =
  K
  2
  K2
  2^74 +2^2058+2^2n =
  (
  2
  37
  )
  2
  +
  2
  2058
  +
  (
  2
  n
  )
  2
  (237)2+22058+(2n)2
  We try to write this expression as
  (
  a
  +
  b
  )
  2
  =
  a
  2
  +
  2
  a
  b
  +
  b
  2
  (a+b)2=a2+2ab+b2
  Now a =
  2
  37
  237, 2ab =
  2
  2058
  22058 and b =
  2
  n
  2n
  Substituting the value of a in 2ab, we get b = 2020
• 6 years agoHelpfull: Yes(0) No(2)