α =

β =

Ανάλυση αριθμών σε γινόμενο πρώτων παραγόντων

ΜΚΔ και ΕΚΠ δύο φυσικών αριθμών (3η μέθοδος)

όλοι οι παράγοντες με τους μεγαλύτερους εκθέτες

κοινοί παράγοντες με τον μικρότερο εκθέτη

Σημειώσεις

α = 2 (slider from 2 to 25000, step 1)
β = 3 (slider from 2 to 25000, step 1)
Lα1 = ΠρώτοιΠαράγοντεςΑριθμού(α)
Lβ1 = ΠρώτοιΠαράγοντεςΑριθμού(β)
Lα2 = Sequence(α / If[n ≟ 0, 1, Product[Lα1, n]], n, 0, Length[Lα1])
Lβ2 = Sequence(β / If[n ≟ 0, 1, Product[Lβ1, n]], n, 0, Length[Lβ1])
Εισ1 = Textfield(α)
Εισ2 = Textfield(β)
Κα2 = TableText({Lα2, Lα1}, "vr|")
Κβ2 = TableText({Lβ2, Lβ1}, "vr|")
Lα3 = Unique(Lα1)
Lβ3 = Unique(Lβ1)
α1 = Element(Lα3, 1)
α2 = Element(Lα3, 2)
α3 = Element(Lα3, 3)
α4 = Element(Lα3, 4)
α5 = Element(Lα3, 5)
α6 = Element(Lα3, 6)
α7 = Element(Lα3, 7)
α8 = Element(Lα3, 8)
α9 = Element(Lα3, 9)
α1ν = Length(KeepIf[x ≟ α1, Lα1])
α2ν = Length(KeepIf[x ≟ α2, Lα1])
α3ν = Length(KeepIf[x ≟ α3, Lα1])
α4ν = Length(KeepIf[x ≟ α4, Lα1])
α5ν = Length(KeepIf[x ≟ α5, Lα1])
α6ν = Length(KeepIf[x ≟ α6, Lα1])
α7ν = Length(KeepIf[x ≟ α7, Lα1])
α8ν = Length(KeepIf[x ≟ α8, Lα1])
α9ν = Length(KeepIf[x ≟ α9, Lα1])
A1: "" + (LaTeX[α]) + " = " + (LaTeX[((α1 * "^") * α1ν)]) + " " + (LaTeX[If[α2 > 1, "·" + (((α2 * "^") * α2ν)), ""]]) + " " + (LaTeX[If[α3 > 1, "·" + (((α3 * "^") * α3ν)), ""]]) + " " + (LaTeX[If[α4 > 1, "·" + (((α4 * "^") * α4ν)), ""]]) + " " + (LaTeX[If[α5 > 1, "·" + (((α5 * "^") * α5ν)), ""]]) + " " + (LaTeX[If[α6 > 1, "·" + (((α6 * "^") * α6ν)), ""]]) + " " + (LaTeX[If[α7 > 1, "·" + (((α7 * "^") * α7ν)), ""]]) + " " + (LaTeX[If[α8 > 1, "·" + (((α8 * "^") * α8ν)), ""]]) + " " + (LaTeX[If[α9 > 1, "·" + (((α9 * "^") * α9ν)), ""]]) + ""
β1 = Element(Lβ3, 1)
β2 = Element(Lβ3, 2)
β3 = Element(Lβ3, 3)
β4 = Element(Lβ3, 4)
β5 = Element(Lβ3, 5)
β6 = Element(Lβ3, 6)
β7 = Element(Lβ3, 7)
β8 = Element(Lβ3, 8)
β9 = Element(Lβ3, 9)
β1ν = Length(KeepIf[x ≟ β1, Lβ1])
β2ν = Length(KeepIf[x ≟ β2, Lβ1])
β3ν = Length(KeepIf[x ≟ β3, Lβ1])
β4ν = Length(KeepIf[x ≟ β4, Lβ1])
β5ν = Length(KeepIf[x ≟ β5, Lβ1])
β6ν = Length(KeepIf[x ≟ β6, Lβ1])
β7ν = Length(KeepIf[x ≟ β7, Lβ1])
β8ν = Length(KeepIf[x ≟ β8, Lβ1])
β9ν = Length(KeepIf[x ≟ β9, Lβ1])
B1: "" + (LaTeX[β]) + "=" + (LaTeX[((β1 * "^") * β1ν)]) + " " + (LaTeX[If[β2 > 1, "·" + (((β2 * "^") * β2ν)), ""]]) + " " + (LaTeX[If[β3 > 1, "·" + (((β3 * "^") * β3ν)), ""]]) + " " + (LaTeX[If[β4 > 1, "·" + (((β4 * "^") * β4ν)), ""]]) + " " + (LaTeX[If[β5 > 1, "·" + (((β5 * "^") * β5ν)), ""]]) + " " + (LaTeX[If[β6 > 1, "·" + (((β6 * "^") * β6ν)), ""]]) + " " + (LaTeX[If[β7 > 1, "·" + (((β7 * "^") * β7ν)), ""]]) + " " + (LaTeX[If[β8 > 1, "·" + (((β8 * "^") * β8ν)), ""]]) + " " + (LaTeX[If[β9 > 1, "·" + (((β9 * "^") * β9ν)), ""]]) + ""
ΑΓΠ = TableText({{A1, B1}}, "{v")
LTomi = Intersection(Lα3, Lβ3)
γ1 = Element(LTomi, 1)
γ2 = Element(LTomi, 2)
γ3 = Element(LTomi, 3)
γ4 = Element(LTomi, 4)
γ5 = Element(LTomi, 5)
γ6 = Element(LTomi, 6)
ΜΚΔ2 = GCD(α, β)
Lγ1 = ΠρώτοιΠαράγοντεςΑριθμού(ΜΚΔ2)
γ1ν = Length(KeepIf[γ1 ≟ x, Lγ1])
γ2ν = Length(KeepIf[γ2 ≟ x, Lγ1])
γ3ν = Length(KeepIf[γ3 ≟ x, Lγ1])
γ4ν = Length(KeepIf[γ4 ≟ x, Lγ1])
γ5ν = Length(KeepIf[γ5 ≟ x, Lγ1])
γ6ν = Length(KeepIf[γ6 ≟ x, Lγ1])
ΕΚΠ2 = LCM(α, β)
Lδ1 = ΠρώτοιΠαράγοντεςΑριθμού(ΕΚΠ2)
LEnosi = Union(Lα1, Lβ1)
δ1 = Element(LEnosi, 1)
δ2 = Element(LEnosi, 2)
δ3 = Element(LEnosi, 3)
δ4 = Element(LEnosi, 4)
δ5 = Element(LEnosi, 5)
δ6 = Element(LEnosi, 6)
δ7 = Element(LEnosi, 7)
δ8 = Element(LEnosi, 8)
δ9 = Element(LEnosi, 9)
δ1ν = Length(KeepIf[δ1 ≟ x, Lδ1])
δ2ν = Length(KeepIf[δ2 ≟ x, Lδ1])
δ3ν = Length(KeepIf[δ3 ≟ x, Lδ1])
δ4ν = Length(KeepIf[δ4 ≟ x, Lδ1])
δ5ν = Length(KeepIf[δ5 ≟ x, Lδ1])
δ6ν = Length(KeepIf[δ6 ≟ x, Lδ1])
δ7ν = Length(KeepIf[δ7 ≟ x, Lδ1])
δ8ν = Length(KeepIf[δ8 ≟ x, Lδ1])
δ9ν = Length(KeepIf[δ9 ≟ x, Lδ1])
ΕΚΠ: "ΕΚΠ" + (LaTeX[LaTeX[(α, β)]]) + "=" + (LaTeX[((δ1 * "^") * δ1ν)]) + " " + (LaTeX[If[δ2 > 1, "·" + (((δ2 * "^") * δ2ν)), ""]]) + " " + (LaTeX[If[δ3 > 1, "·" + (((δ3 * "^") * δ3ν)), ""]]) + " " + (LaTeX[If[δ4 > 1, "·" + (((δ4 * "^") * δ4ν)), ""]]) + " " + (LaTeX[If[δ5 > 1, "·" + (((δ5 * "^") * δ5ν)), ""]]) + " " + (LaTeX[If[δ6 > 1, "·" + (((δ6 * "^") * δ6ν)), ""]]) + " " + (LaTeX[If[δ7 > 1, "·" + (((δ7 * "^") * δ7ν)), ""]]) + " " + (LaTeX[If[δ8 > 1, "·" + (((δ8 * "^") * δ8ν)), ""]]) + " " + (LaTeX[If[δ9 > 1, "·" + (((δ9 * "^") * δ9ν)), ""]]) + "=" + (LaTeX[ΕΚΠ2]) + ""
Α = (3.599254807692308, 4.091850961538461)
Β = (7.924495192307689, 4.091850961538461)
a = false
b = false
κείμενο3: "ΜΚΔ " + (LaTeX[LaTeX[(α, β)]]) + "="
κείμενο4: "ΕΚΠ" + (LaTeX[LaTeX[(α, β)]]) + "="
ΜΚΔ: "ΜΚΔ" + (LaTeX[LaTeX[(α, β)]]) + "  " + (LaTeX[If[γ1 > 1, "=" + (((γ1 * "^") * γ1ν)), ""]]) + " " + (LaTeX[If[γ2 > 1, "·" + (((γ2 * "^") * γ2ν)), ""]]) + " " + (LaTeX[If[γ3 > 1, "·" + (((γ3 * "^") * γ3ν)), ""]]) + " " + (LaTeX[If[γ4 > 1, "·" + (((γ4 * "^") * γ4ν)), ""]]) + " " + (LaTeX[If[γ5 > 1, "·" + (((γ5 * "^") * γ5ν)), ""]]) + " " + (LaTeX[If[γ6 > 1, "·" + (((γ6 * "^") * γ6ν)), ""]]) + "=" + (LaTeX[ΜΚΔ2]) + ""
c = false
κείμενο5: "Οι αριθμοί  " + α + " και " + β + "  είναι
  πρώτοι μεταξύ τους"
e = false
κείμενο8: "Ο αριθμός " + α + "  
είναι πρώτος"
κείμενο9: "Ο αριθμός " + β + "
είναι πρώτος"
Γ = (2.7114423076923115, 0.03978365384615287)
Δ = (2.6886778846153883, -0.8707932692307703)

Tool: Πρώτοι Παράγοντες Αριθμού (command ΠρώτοιΠαράγοντεςΑριθμού)
Δώστε ένα δρομέα α με τιμές μεγαλύτερες του 2 ή έναν αριθμό β
Input: a
Output: κα1

λ11 = Sequence(If[Mod[a, k] ≟ 0, Div[a, k], 0], k, 1, a, 1)
λ12 = Sort(λ11)
λ13 = KeepIf((x ≥ 1) ∧ (x ≤ a), λ12)
μ1 = Length(λ13)
α_1 = Element(λ13, 2)
β_1 = Element(λ13, μ1 - 1)
λ21 = Sequence(If[Mod[β_1, k] ≟ 0, Div[β_1, k], 0], k, 1, β_1, 1)
λ22 = Sort(λ21)
λ23 = KeepIf((x ≥ 1) ∧ (x ≤ β_1), λ22)
α_2 = Element(λ23, 2)
μ2 = Length(λ23)
β_2 = Element(λ23, μ2 - 1)
λ31 = Sequence(If[Mod[β_2, k] ≟ 0, Div[β_2, k], 0], k, 1, β_2, 1)
λ32 = Sort(λ31)
λ33 = KeepIf((x ≥ 1) ∧ (x ≤ β_2), λ32)
α_3 = Element(λ33, 2)
μ3 = Length(λ33)
β_3 = Element(λ33, μ3 - 1)
λ41 = Sequence(If[Mod[β_3, k] ≟ 0, Div[β_3, k], 0], k, 1, β_3, 1)
λ42 = Sort(λ41)
λ43 = KeepIf((x ≥ 1) ∧ (x ≤ β_3), λ42)
μ4 = Length(λ43)
α_4 = Element(λ43, 2)
β_4 = Element(λ43, μ4 - 1)
λ51 = Sequence(If[Mod[β_4, k] ≟ 0, Div[β_4, k], 0], k, 1, β_4, 1)
λ52 = Sort(λ51)
λ53 = KeepIf((x ≥ 1) ∧ (x ≤ β_4), λ52)
μ5 = Length(λ53)
α_5 = Element(λ53, 2)
β_5 = Element(λ53, μ5 - 1)
λ61 = Sequence(If[Mod[β_5, k] ≟ 0, Div[β_5, k], 0], k, 1, β_5, 1)
λ62 = Sort(λ61)
λ63 = KeepIf((x ≥ 1) ∧ (x ≤ β_5), λ62)
μ6 = Length(λ63)
α_6 = Element(λ63, 2)
β_6 = Element(λ63, μ6 - 1)
λ71 = Sequence(If[Mod[β_6, k] ≟ 0, Div[β_6, k], 0], k, 1, β_6, 1)
λ72 = Sort(λ71)
λ73 = KeepIf((x ≥ 1) ∧ (x ≤ β_6), λ72)
μ7 = Length(λ73)
α_7 = Element(λ73, 2)
β_7 = Element(λ73, μ7 - 1)
λ81 = Sequence(If[Mod[β_7, k] ≟ 0, Div[β_7, k], 0], k, 1, β_7, 1)
λ82 = Sort(λ81)
λ83 = KeepIf((x ≥ 1) ∧ (x ≤ β_7), λ82)
α_8 = Element(λ83, 2)
κα: {α_1, α_2, α_3, α_4, α_5, α_6, α_7, α_8}
κα1 = RemoveUndefined(κα)