Basic Formulas:
1. (a + b)(a  b) = (a^{2}  b^{2})
2. (a + b)^{2} = (a^{2} + b^{2} + 2ab)
3. (a  b)^{2} = (a^{2} + b^{2}  2ab)
4. (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)
5. (a^{3} + b^{3}) = (a + b)(a^{2}  ab + b^{2})
6. (a^{3}  b^{3}) = (a  b)(a^{2} + ab + b^{2})
7. (a^{3} + b^{3} + c^{3}  3abc) = (a + b + c)(a^{2} + b^{2} + c^{2}  ab  bc  ac)
8. When a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc.
1. 

Let, the number be X. Then
X – 36 = 86 – X
2X = 122
X = 61
2. 

Let, the number be X. Then
7X – 15 = 2X + 10
5X = 25
X = 5
3. 

Let the number be X.
Then, X + 1/X = 13/6 ⇔(X² + 1)/X = 13/6 ⇔6X²  13X + 6 = 0
⇒ 6X²  9X – 4X +6 = 0
⇒ (3X  2)(2X  3) = 0
⇒ X = 2/3 or 3/2
Hence, the required number is 2/3 or 3/2
4. 

Let the numbers be X and (184  X). Then,
⇒ X/3 – (184  X)/7 = 8
⇒ 7X – 3(184  X) = 168
⇒ 10X = 720
⇒ X = 72
5. 

Let the numbers be X and (15  X)
Then, X² + (15  X)² = 113
⇒ X² + 225 + X²  30X = 113
⇒ 2X²  30X + 112 = 0
⇒ X² 15X + 56 = 0
⇒ (X  7)(X  8) = 0
⇒ X = 7 or X = 8
So, the numbers are 7 and 8.
6. 

Let, the consecutive even numbers be X, X+2, X+4 and X+6.
Then, sum of these numbers = (27 x 4) = 108
So, X + X + 2 + X + 4 + X + 6 = 108
4X = 96
X = 24
∴ Largest number = (X + 6) = 30
7. 

Let, the ten's digit be X. then unit's digit = (X + 3).
Sum of digits = X + (X + 3) = 2X + 3
Number = 10X + (X + 3) = 11X + 3
∴ (11X + 3) ÷ (2X + 3) = 4/1
⇒ 11 X + 3 = 4(2 X +3)
⇒ 3X =9
⇒ X = 3
Hence, Required number = 11 X + 3 = 36
8. 

Let, the two parts be X and (50  X).
Then, 1/X + 1/(50X) = 1/12
⇒ (50 – X + X) ÷ X(50 – X) = 1/12
⇒ X²  50X + 600 = 0
⇒ X²  30X – 20X + 600 = 0
⇒ (X  30) (X  20) = 0
⇒ X = 30 or X = 20
So, the parts are 30 and 20.
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