Answer to Question #350945 in Combinatorics | Number Theory for WSN

Question #350945

2. Find all integers x #= 3 such that x-3Ix3-3.


1
Expert's answer
2022-06-27T09:34:36-0400

"\\dfrac {x^3-3}{x-3}=~ \\dfrac {x^3-27+24}{x-3}= ~\\dfrac {(x-3)(x\u00b2+3x+9)+24}{x-3}=~\\underbrace{x\u00b2+3x+9}_{\\text{integer}}~+~\\dfrac {24}{x-3}"


So, we are looking for values of x, such that:


"x-3 ~|~24 \\implies \\\\\n\\implies ~x-~3\\in~\\begin{Bmatrix}\n \u00b11,\u00b12,\u00b13,\u00b14,\u00b16,\u00b18,\u00b112,\u00b124\n\\end{Bmatrix} \\implies \\\\\n\\implies ~x \\in ~ \\begin{Bmatrix}\n\u221221,\u22129,\u22125,-3,\u22121,0,1,2,4,5,6,7,9,11,15,27\n\\end{Bmatrix}"



Corresponding x values are


"\u221221,\u22129,\u22125,\u22123,\u22121,0,1,2,4,5,6,7,9,11,15 ~and~ 27"





















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