Find the values of n for which n+18 and n+90 is a perfect square? ?

• n+18 = y^2 & n+90 = x^2
  
  then (x^2-y^2) = 72
  => (x+y)*(x-y) = 72*1 or 36*2 or 24*3 or 18*4 or 12*6 or 9*8
  
  solving we get 6 solns
  2 +ve soln n=31,271
  1 -ve soln n = -9
  3 decimal soln n = 92.25 , ... , ....
  
• 9 years agoHelpfull: Yes(33) No(3)
• 2 solutions..
  For n= 31,
  n + 18 = 31+18 = 49
  n + 90 = 31+90 = 121, both are perfect square
  
  For n= 271,
  n+ 18 = 271 + 18 = 289
  n+ 90 = 271 + 90 = 361, both are again perfect squares
• 9 years agoHelpfull: Yes(13) No(8)
• I have find 3 such case.
  
  n = -9, 31 and 271
  
  Put n= -9
  (n+18)= (-9+18)
  = 9 which is a perfect square
  (n+90)= (-9+90)
  = 81 which is a perfect square
  
  Put n= 31
  (n+18)= (31+18)
  = 49 which is a perfect square
  (n+90)= (31+90)
  = 121 which is a perfect square
  
  Put n= 271
  (n+18)= (271+18)
  = 289 which is a perfect square
  (n+90)= (271+90)
  = 361 which is a perfect square
• 9 years agoHelpfull: Yes(8) No(2)
• let, a^2=n+18
  b^2=n+90
  b^2=n+18+72
  b^2-a^2=72
  therefore,
  (b-a)(b+a)=72
  so, 1x72 solving equ. b-a and b+a it give fractional value of a and b
  2x36 solving equ. b-a and b+a it give int value of a and b(solution)
  3x24 solving equ. b-a and b+a it give fractional value of a and b
  4x18 solving equ. b-a and b+a it give int value of a and b(solution)
  6x12 solving equ. b-a and b+a it give int value of a and b(solution)
  8x9 solving equ. b-a and b+a it give fractional value of a and b
  
  therefore , there are only 3 solution
  
• 9 years agoHelpfull: Yes(7) No(0)
• 31+90=121=11^2
  31+18=49=7^2
  hence 31 ans
  by hit and trial method
• 9 years agoHelpfull: Yes(3) No(2)
• here I marked option 2 i.e only 2 Solution..once result will come then only I ll come to know whose ans. is correct.
• 9 years agoHelpfull: Yes(3) No(0)
• Options are:- 1,2,4, none
• 9 years agoHelpfull: Yes(2) No(0)
• See the trick
  IF a^2 - b^2 = ODD
  Thn find the factor of that no devide it by 2 . that will be ans
  IF a^2 - b^2 = EVEN
  Then if the no is not divisible by 4 then no sol
  If it divisible by 4 then devide that no by 4
  find the factors and then devide the factor by 2 that will be ans
  
  
  
  HEre
  n+18 = b^2 an n+90 = a^2
  a^2 - b^2 = 72
  now 72/4 = 18
  18= 2*9*9
  so factors are 2*3 = 6
  so 6/2 = 3 Factors
  
  
• 9 years agoHelpfull: Yes(1) No(0)
• qustion asked that x,y,n should be positive,whole no so only two solution.
  option was
  1.1
  2.2
  3.4
  4.none
• 9 years agoHelpfull: Yes(1) No(0)
• n=31
  answer is 1.
• 9 years agoHelpfull: Yes(0) No(3)
• is there any method to solve this question or else only trail nd error ??? mr. ABHINAV ARPANAM
• 9 years agoHelpfull: Yes(0) No(0)
• I used hit and trial method in exam, so basically I don't no any other method
• 9 years agoHelpfull: Yes(0) No(0)
• 31
  obviously...
  
• 9 years agoHelpfull: Yes(0) No(1)
• This question was asked today 30 august
  I too got the ans-31,271,-9
  
• 9 years agoHelpfull: Yes(0) No(0)
• n+18 = a^2
  n+90 = b^2
  subtract the two
  b^2 - a^2 = 72
  (b-a)(b+a) = 72
  Look at factors of 72
  1*72
  2*36
  3*24
  4*18
  6*12
  8*9
  
  How many of these have an a and b such that (b-a) is the small number and (b+a) is the large?
  b-a = 1
  b+a = 72
  add the two
  2b = 73 (no... needs to be whole numbers)
  b-a = 2
  b+a = 36
  2b = 38
  b = 19, a = 17, b^2 = 361, n + 90 = 361 so n = 271
  n should also equal 17^2 - 18 = 271 (so that's 1)
  
  we notice the sum of the factors must be even and the average of the factors is b
  
  3*24 sum is 27
  4*18 sum is 22, the average is 11
  b = 11
  n = 11^2 - 90 = 31
  a = b - small factor = 11-4 = 7
  check : 31+18 = 49 = 7^2 (so that's 2)
  
  6*12 sum is 18, average is 9
  b = 9, a = 9-6 = 3
  n= 9^2 - 90 = -9
  -9 + 18 = 9 = 3^2 (that's 3)
  
• 9 years agoHelpfull: Yes(0) No(0)
• Can any body remembered the exact option of this Question.
• 9 years agoHelpfull: Yes(0) No(0)
• n+18
  n+90
  by guessing we can tell that n=31
  as 31+18=49 sqrt(7)
  and 31+90=121 sqrt(11)
• 9 years agoHelpfull: Yes(0) No(0)
• 31+18 = 49 (p.sq.) & 31+90 = 121 (p.sq.)
  n = 31 ans
• 9 years agoHelpfull: Yes(0) No(0)
• ind the values of n for which n+18 and n+90 is a perfect square ?
  A)
  6
  
  B)
  5
  
  C)
  4
  
  D)
  3
  
  
  Like (2) - Dislike (0) - Comments (0)
  1 Explanation
  
  1
  
  Kunal Kumar Deewan2 years ago edited
  1 upvotes
  
  Answer : C
  Explanation :
  Let, n+18 = a2 and n+90 = b2
  
  n+18 = a2
  
  => n = (a2 - 18) ------------------ (1)
  
  => n = (b2 - 90) ------------------ (2)
  
  from (1) and (2):
  
  (a2 - 18) = (b2 - 90)
  
  => b2 - a2 = 72
  
  => (b+a)(b-a) = 72
  
  => (b+a)(b-a) = 36x2 OR 24x3 OR 18x4 OR 12x6 OR 9x8
  
  Case 1:
  
  (b+a)(b-a) = 36x2
  
  Possible value of a and b are: 17 and 19
  
  In this case, n = (a2 - 18) => n = 271
  
  Case 2:
  
  (b+a)(b-a) = 24x3
  
  Possible value of a and b are: 10.5 and 13.5
  
  In this case, n = (a2 - 18) => n = 92.25
  
  Case 3:
  
  (b+a)(b-a) = 18x4
  
  Possible value of a and b are: 7 and 11
  
  In this case, n = (a2 - 18) => n = 31
  
  Case 4:
  
  (b+a)(b-a) = 12x6
  
  Possible value of a and b are: 3 and 9
  
  In this case, n = (a2 - 18) => n = -9
  
  Case 5:
  
  (b+a)(b-a) = 9x8
  
  No real values of a and b satisfy this equation
  
  So, there are total 4 possible values of n
• 7 years agoHelpfull: Yes(0) No(0)