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Question Number 223348 by cryptograph last updated on 21/Jul/25
Determine gcd(13a+19b,ab) given that gcd(a,19)=gcd(b,13)=1
$${Determine}\:{gcd}\left(\mathrm{13}{a}+\mathrm{19}{b},{ab}\right)\:{given}\:{that}\:{gcd}\left({a},\mathrm{19}\right)={gcd}\left({b},\mathrm{13}\right)=\mathrm{1} \\ $$
Commented by A5T last updated on 23/Jul/25
This isn′t unique. a=1 and b=1⇒ gcd(32,1)=1 a=2 and b=2 ⇒ gcd(64,4)=4
$$\mathrm{This}\:\mathrm{isn}'\mathrm{t}\:\mathrm{unique}.\:\mathrm{a}=\mathrm{1}\:\mathrm{and}\:\mathrm{b}=\mathrm{1}\Rightarrow\:\mathrm{gcd}\left(\mathrm{32},\mathrm{1}\right)=\mathrm{1} \\ $$$$\mathrm{a}=\mathrm{2}\:\mathrm{and}\:\mathrm{b}=\mathrm{2}\:\Rightarrow\:\mathrm{gcd}\left(\mathrm{64},\mathrm{4}\right)=\mathrm{4} \\ $$
Answered by Raphael254 last updated on 24/Jul/25
a ≠ 19p, p ∈ Z^∗ b ≠ 13q, q ∈ Z^∗ gcd(13a + 19b, ab) if ∃k ∈ Z^∗ ⇒ ab = k(13a + 19b), k = ((ab)/(13 a + 19b)), gcd(13a + 19b, ab) = ∣13 a + 19b∣, if not, gcd(13a + 19b, ab) = 1 ((ab)/(13a + 19b)) = (b/(13)) − ((19b^2 )/(13^2 a)) + ((19^2 b^3 )/(13^3 a^2 )) − ... ± ((19^(n−1) b^n )/(13^n a^(n−1) )), n ∈ N^∗ and means number of terms. There is something interesting in this; if a = 19p and b = 13q: ((19p×13q)/(13×19p + 19×13q)) = ((19×13(pq))/(19×13(p + q))) = ((pq)/(p + q)) = ((p(q))/(p(1 + (q/p)))) = (q/( 1 + (q/p))) q = k(1 + (q/p)) ⇔ ((pq − kp − kq)/p) = 0 ⇒ q − k − ((kq)/p) = 0 Per example: (p ≤ q and q multiple of p) p = 2, q = 6 6 − k − ((6k)/2) = 0 −4k = −6 k = (3/2)∉ Z^∗ p = 3, q = 6 6 − k − ((6k)/3) = 0 −3k = −6 k = 2 Proof: q = k(1 + (q/p)) 6 = 2(1 + (6/3)) 6 = 6 a =19p and b = 13q a = 19×3 = 57 b = 13×6 = 78 13a + 19b = 13×57 + 19×78 ab = 57×78 k = ((57×78)/(13×57 + 19×78)) = 2 It means that the same k we found before is the same of this time, so: But how we are considering a ≠ 19p and b ≠ 13q, it is not necessary to calculate it. if a = b: (a^2 /(13a + 19a)) = (a/(13 + 19)) = (a/(32)), here we can see that the only values of a possible is a = 32k, k ∈ Z^∗ . In this case, gcd(13a + 19b, ab) = ∣13(32k) + 19(32k)∣ = ∣416k + 608k∣ = ∣1024k∣ or ∣(2^(10) )k∣ See that there are infinites gcds, when ab ≥ 13b + 19b. So doesn′t exist a maximum gcd in this case. But anyway: (a/(32)) = (a/(13)) − ((19a^2 )/(13^2 a)) + ((19^2 a^3 )/(13^3 a^2 )) − ... ± ((19^(n−1) a^n )/(13^n a^(n−1) )), n ∈ N^∗ (a/(32)) = (a/(13)) − ((19a)/(13^2 )) + ((19^2 a)/(13^3 )) − ... ± ((19^(n−1) a)/(13^n )), n ∈ N^∗ See that the a stays in the numerator for all the sum, so: (a/(13)) − ((19a)/(13^2 )) = ((13a − 19a)/(13^2 )) = −((6a)/(13^2 )) −((6a)/(13^2 )) + ((19^2 a)/(13^3 )) = ((−6a×13 + 19^2 a)/(13^3 )) = ((283a)/(13^3 )) It seems strange the way it oscilates, but we can say: Σ_(n = 1) ^∞ ((19^(n−1) a^n )/(13^n b^(n−1) )) = ((ab)/(13a + 19b)) Σ_(n = 1) ^∞ ((19^(n−1) a)/(13^n )) = (a/(32)) if a = mb, m ∈ Z^∗ : ((mb^2 )/(13mb + 19b)) = ((mb)/(13m + 19)) if b = na, n ∈ Z^∗ : ((a^2 n)/(13a + 19na)) = ((an)/(19n + 13)) if a ≠ mb, a = mb + r; mb, r ∈ Z^∗ : (((mb + r)b)/(13(mb + r) + 19b)) if b ≠ na, b = na + r; na, r ∈ Z^∗ : ((a(na + r))/(13a + 19(na + r))) if ∃k ∈ Z^∗ ⇒ 13a + 19b= k(ab), k = ((13a + 19b)/(ab)), gcd(13a + 19b, ab) = ∣ab∣, if not, gcd(13a + 19b, ab) = 1 ((13a + 19b)/(ab)) = ((13)/b) + ((19)/a) If a = b: ((13 a + 19 a)/a^2 ) = ((13 + 19)/a) = ((32)/a); here the fraction inverted, and a = ±1, ±2, ±4, ±8, ±16 or ±32 to make k a integer Maximum gdc on this case = ∣a^2 ∣ = ∣(±32)^2 ∣ = 1024 Arbitrary a,b ∈ Z^∗ gdc; if a = mb, m ∈ Z^∗ : ((13kb + 19b)/(mb^2 )) = ((b(13m + 19))/(mb^2 )) = ((13m + 19)/(mb)) = ((13)/b) + ((19)/(mb)) gcd = ∣ab∣; a, b solutions to the above sum resulting in k ∈ Z^∗ if b = na, n ∈ Z^∗ : ((13a + 19na)/(na^2 )) = ((a(13 + 19n))/(na^2 )) = ((13 + 19n)/(na)) = ((19)/a) + ((13)/(na)) gcd = ∣ab∣; a, b solutions to the above sum resulting in k ∈ Z^∗ if a ≠ mb, a = mb + r; mb, r ∈ Z^∗ : ((13(mb + r) + 19b)/((mb + r)b)) ((13)/b) + ((19)/(mb + r)) if b ≠ na, b = na + r; na, r ∈ Z^∗ : ((13a + 19(na + r))/(a(na + r))) ((19)/a) + ((13)/(na + r)) all possible gcds when ab and 13a + 19b are multiple of each other: if ab ≥ 13a + 19b: k = ((mb)/(13m + 19)) ; ((na)/(19n − 13)) k = (a/(19)) + ((na)/(13)) ; k = (((mb + x)b)/(13(mb + x) + 19b)) ; k = ((a(na + x))/(13a + 19(na + x))), with (a, b ∈ Z^∗ ), (m, n ∈ Q^∗ ) HH
$${a}\:\neq\:\mathrm{19}{p},\:{p}\:\in\:\mathbb{Z}^{\ast} \\ $$$${b}\:\neq\:\mathrm{13}{q},\:{q}\:\in\:\mathbb{Z}^{\ast} \\ $$$$ \\ $$$${gcd}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b},\:{ab}\right) \\ $$$$ \\ $$$${if}\:\exists{k}\:\in\:\mathbb{Z}^{\ast} \:\Rightarrow\:{ab}\:=\:{k}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b}\right), \\ $$$${k}\:=\:\frac{{ab}}{\mathrm{13}\:{a}\:+\:\mathrm{19}{b}},\:{gcd}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b},\:{ab}\right)\:=\:\mid\mathrm{13}\:{a}\:+\:\mathrm{19}{b}\mid, \\ $$$${if}\:{not},\:{gcd}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b},\:{ab}\right)\:=\:\mathrm{1} \\ $$$$ \\ $$$$\frac{{ab}}{\mathrm{13}{a}\:+\:\mathrm{19}{b}}\:=\:\frac{{b}}{\mathrm{13}}\:−\:\frac{\mathrm{19}{b}^{\mathrm{2}} }{\mathrm{13}^{\mathrm{2}} {a}}\:+\:\frac{\mathrm{19}^{\mathrm{2}} {b}^{\mathrm{3}} }{\mathrm{13}^{\mathrm{3}} {a}^{\mathrm{2}} }\:−\:…\:\pm\:\frac{\mathrm{19}^{{n}−\mathrm{1}} {b}^{{n}} }{\mathrm{13}^{{n}} {a}^{{n}−\mathrm{1}} },\:{n}\:\in\:\mathbb{N}^{\ast} \:{and}\:{means}\:{number}\:{of}\:{terms}. \\ $$$$ \\ $$$${There}\:{is}\:{something}\:{interesting}\:{in}\:{this};\:{if}\:{a}\:=\:\mathrm{19}{p}\:{and}\:{b}\:=\:\mathrm{13}{q}: \\ $$$$ \\ $$$$\frac{\mathrm{19}{p}×\mathrm{13}{q}}{\mathrm{13}×\mathrm{19}{p}\:+\:\mathrm{19}×\mathrm{13}{q}}\:=\:\frac{\mathrm{19}×\mathrm{13}\left({pq}\right)}{\mathrm{19}×\mathrm{13}\left({p}\:+\:{q}\right)}\:=\:\frac{{pq}}{{p}\:+\:{q}}\:=\:\frac{{p}\left({q}\right)}{{p}\left(\mathrm{1}\:+\:\frac{{q}}{{p}}\right)}\:\:=\:\frac{{q}}{\:\mathrm{1}\:+\:\frac{{q}}{{p}}} \\ $$$$ \\ $$$${q}\:=\:{k}\left(\mathrm{1}\:+\:\frac{{q}}{{p}}\right)\:\Leftrightarrow\:\frac{{pq}\:−\:{kp}\:−\:{kq}}{{p}}\:=\:\mathrm{0}\:\Rightarrow\:{q}\:−\:{k}\:−\:\frac{{kq}}{{p}}\:=\:\mathrm{0} \\ $$$$ \\ $$$${Per}\:{example}:\:\left({p}\:\leq\:{q}\:{and}\:{q}\:{multiple}\:{of}\:{p}\right) \\ $$$$ \\ $$$${p}\:=\:\mathrm{2},\:{q}\:=\:\mathrm{6} \\ $$$$ \\ $$$$\mathrm{6}\:−\:{k}\:−\:\frac{\mathrm{6}{k}}{\mathrm{2}}\:=\:\mathrm{0} \\ $$$$−\mathrm{4}{k}\:=\:−\mathrm{6} \\ $$$${k}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\notin\:\mathbb{Z}^{\ast} \\ $$$$ \\ $$$${p}\:=\:\mathrm{3},\:{q}\:=\:\mathrm{6} \\ $$$$ \\ $$$$\mathrm{6}\:−\:{k}\:−\:\frac{\mathrm{6}{k}}{\mathrm{3}}\:=\:\mathrm{0} \\ $$$$−\mathrm{3}{k}\:=\:−\mathrm{6} \\ $$$${k}\:=\:\mathrm{2} \\ $$$$ \\ $$$${Proof}: \\ $$$$ \\ $$$${q}\:=\:{k}\left(\mathrm{1}\:+\:\frac{{q}}{{p}}\right) \\ $$$$\mathrm{6}\:=\:\mathrm{2}\left(\mathrm{1}\:+\:\frac{\mathrm{6}}{\mathrm{3}}\right) \\ $$$$\mathrm{6}\:=\:\mathrm{6} \\ $$$$ \\ $$$${a}\:=\mathrm{19}{p}\:{and}\:{b}\:=\:\mathrm{13}{q} \\ $$$${a}\:=\:\mathrm{19}×\mathrm{3}\:=\:\mathrm{57} \\ $$$${b}\:=\:\mathrm{13}×\mathrm{6}\:=\:\mathrm{78} \\ $$$$ \\ $$$$\mathrm{13}{a}\:+\:\mathrm{19}{b}\:=\:\mathrm{13}×\mathrm{57}\:+\:\mathrm{19}×\mathrm{78} \\ $$$${ab}\:=\:\mathrm{57}×\mathrm{78} \\ $$$$ \\ $$$${k}\:=\:\frac{\mathrm{57}×\mathrm{78}}{\mathrm{13}×\mathrm{57}\:+\:\mathrm{19}×\mathrm{78}}\:=\:\mathrm{2} \\ $$$$ \\ $$$${It}\:{means}\:{that}\:{the}\:{same}\:{k}\:{we}\:{found}\:{before}\:{is}\:{the}\:{same}\:{of}\:{this}\:{time},\:{so}: \\ $$$${But}\:{how}\:{we}\:{are}\:{considering}\:{a}\:\neq\:\mathrm{19}{p}\:{and}\:{b}\:\neq\:\mathrm{13}{q},\:{it}\:{is}\:{not}\:{necessary}\:{to}\:{calculate}\:{it}. \\ $$$$ \\ $$$${if}\:{a}\:=\:{b}: \\ $$$$ \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{13}{a}\:+\:\mathrm{19}{a}}\:=\:\frac{{a}}{\mathrm{13}\:+\:\mathrm{19}}\:=\:\frac{{a}}{\mathrm{32}},\:{here}\:{we}\:{can}\:{see}\:{that}\:{the}\:{only}\:{values}\:{of}\:\boldsymbol{{a}}\:{possible}\:{is}\:\boldsymbol{{a}}\:=\:\mathrm{32}{k},\:{k}\:\in\:\mathbb{Z}^{\ast} .\:{In}\:{this}\:{case},\:{gcd}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b},\:{ab}\right)\:=\:\mid\mathrm{13}\left(\mathrm{32}{k}\right)\:+\:\mathrm{19}\left(\mathrm{32}{k}\right)\mid\:=\:\mid\mathrm{416}{k}\:+\:\mathrm{608}{k}\mid\:=\:\mid\mathrm{1024}{k}\mid\:{or}\:\mid\left(\mathrm{2}^{\mathrm{10}} \right){k}\mid \\ $$$$ \\ $$$${See}\:{that}\:{there}\:{are}\:{infinites}\:{gcds},\:{when}\:{ab}\:\geq\:\mathrm{13}{b}\:+\:\mathrm{19}{b}.\:{So}\:{doesn}'{t}\:{exist}\:{a}\:{maximum}\:{gcd}\:{in}\:{this}\:{case}. \\ $$$$ \\ $$$${But}\:{anyway}: \\ $$$$ \\ $$$$\frac{{a}}{\mathrm{32}}\:=\:\frac{{a}}{\mathrm{13}}\:−\:\frac{\mathrm{19}{a}^{\mathrm{2}} }{\mathrm{13}^{\mathrm{2}} {a}}\:+\:\frac{\mathrm{19}^{\mathrm{2}} {a}^{\mathrm{3}} }{\mathrm{13}^{\mathrm{3}} {a}^{\mathrm{2}} }\:−\:…\:\pm\:\frac{\mathrm{19}^{{n}−\mathrm{1}} {a}^{{n}} }{\mathrm{13}^{{n}} {a}^{{n}−\mathrm{1}} },\:{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$\frac{{a}}{\mathrm{32}}\:=\:\frac{{a}}{\mathrm{13}}\:−\:\frac{\mathrm{19}{a}}{\mathrm{13}^{\mathrm{2}} }\:+\:\frac{\mathrm{19}^{\mathrm{2}} {a}}{\mathrm{13}^{\mathrm{3}} }\:−\:…\:\pm\:\frac{\mathrm{19}^{{n}−\mathrm{1}} {a}}{\mathrm{13}^{{n}} },\:{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$ \\ $$$${See}\:{that}\:{the}\:\boldsymbol{{a}}\:{stays}\:{in}\:{the}\:{numerator}\:{for}\:{all}\:{the}\:{sum},\:{so}: \\ $$$$ \\ $$$$\frac{{a}}{\mathrm{13}}\:−\:\frac{\mathrm{19}{a}}{\mathrm{13}^{\mathrm{2}} }\:=\:\frac{\mathrm{13}{a}\:−\:\mathrm{19}{a}}{\mathrm{13}^{\mathrm{2}} }\:=\:−\frac{\mathrm{6}{a}}{\mathrm{13}^{\mathrm{2}} } \\ $$$$−\frac{\mathrm{6}{a}}{\mathrm{13}^{\mathrm{2}} }\:+\:\frac{\mathrm{19}^{\mathrm{2}} {a}}{\mathrm{13}^{\mathrm{3}} }\:=\:\frac{−\mathrm{6}{a}×\mathrm{13}\:+\:\mathrm{19}^{\mathrm{2}} {a}}{\mathrm{13}^{\mathrm{3}} }\:=\:\frac{\mathrm{283}{a}}{\mathrm{13}^{\mathrm{3}} } \\ $$$$ \\ $$$${It}\:{seems}\:{strange}\:{the}\:{way}\:{it}\:{oscilates},\:{but}\:{we}\:{can}\:{say}: \\ $$$$ \\ $$$$\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{19}^{{n}−\mathrm{1}} {a}^{{n}} }{\mathrm{13}^{{n}} {b}^{{n}−\mathrm{1}} }\:=\:\frac{{ab}}{\mathrm{13}{a}\:+\:\mathrm{19}{b}} \\ $$$$\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{19}^{{n}−\mathrm{1}} {a}}{\mathrm{13}^{{n}} }\:=\:\frac{{a}}{\mathrm{32}} \\ $$$$ \\ $$$${if}\:{a}\:=\:{mb},\:{m}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{{mb}^{\mathrm{2}} }{\mathrm{13}{mb}\:+\:\mathrm{19}{b}}\:=\:\frac{{mb}}{\mathrm{13}{m}\:+\:\mathrm{19}} \\ $$$$ \\ $$$${if}\:{b}\:=\:{na},\:{n}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{{a}^{\mathrm{2}} {n}}{\mathrm{13}{a}\:+\:\mathrm{19}{na}}\:=\:\frac{{an}}{\mathrm{19}{n}\:+\:\mathrm{13}} \\ $$$$ \\ $$$${if}\:{a}\:\neq\:{mb},\:{a}\:=\:{mb}\:+\:{r};\:{mb},\:{r}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{\left({mb}\:+\:{r}\right){b}}{\mathrm{13}\left({mb}\:+\:{r}\right)\:+\:\mathrm{19}{b}} \\ $$$$ \\ $$$${if}\:{b}\:\neq\:{na},\:{b}\:=\:{na}\:+\:{r};\:{na},\:{r}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{{a}\left({na}\:+\:{r}\right)}{\mathrm{13}{a}\:+\:\mathrm{19}\left({na}\:+\:{r}\right)} \\ $$$$ \\ $$$${if}\:\exists{k}\:\in\:\mathbb{Z}^{\ast} \:\Rightarrow\:\mathrm{13}{a}\:+\:\mathrm{19}{b}=\:{k}\left({ab}\right), \\ $$$${k}\:=\:\frac{\mathrm{13}{a}\:+\:\mathrm{19}{b}}{{ab}},\:{gcd}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b},\:{ab}\right)\:=\:\mid{ab}\mid, \\ $$$${if}\:{not},\:{gcd}\left(\mathrm{13}{a}\:+\:\mathrm{19}{b},\:{ab}\right)\:=\:\mathrm{1} \\ $$$$ \\ $$$$\frac{\mathrm{13}{a}\:+\:\mathrm{19}{b}}{{ab}}\:=\:\frac{\mathrm{13}}{{b}}\:+\:\frac{\mathrm{19}}{{a}} \\ $$$$ \\ $$$${If}\:{a}\:=\:{b}: \\ $$$$ \\ $$$$\frac{\mathrm{13}\:{a}\:+\:\mathrm{19}\:{a}}{{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{13}\:+\:\mathrm{19}}{{a}}\:=\:\frac{\mathrm{32}}{{a}};\:{here}\:{the}\:{fraction}\:{inverted},\:{and}\:{a}\:=\:\pm\mathrm{1},\:\pm\mathrm{2},\:\pm\mathrm{4},\:\pm\mathrm{8},\:\pm\mathrm{16}\:{or}\:\pm\mathrm{32}\:{to}\:{make}\:\boldsymbol{{k}}\:{a}\:{integer} \\ $$$$ \\ $$$${Maximum}\:{gdc}\:{on}\:{this}\:\boldsymbol{{c}}{ase}\:=\:\mid{a}^{\mathrm{2}} \mid\:=\:\mid\left(\pm\mathrm{32}\right)^{\mathrm{2}} \mid\:=\:\mathrm{1024} \\ $$$$ \\ $$$${Arbitrary}\:{a},{b}\:\in\:\mathbb{Z}^{\ast} \:{gdc};\:{if}\:{a}\:=\:{mb},\:{m}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{\mathrm{13}{kb}\:+\:\mathrm{19}{b}}{{mb}^{\mathrm{2}} }\:=\:\frac{{b}\left(\mathrm{13}{m}\:+\:\mathrm{19}\right)}{{mb}^{\mathrm{2}} }\:=\:\frac{\mathrm{13}{m}\:+\:\mathrm{19}}{{mb}}\:=\:\frac{\mathrm{13}}{{b}}\:+\:\frac{\mathrm{19}}{{mb}} \\ $$$$ \\ $$$${gcd}\:=\:\mid{ab}\mid;\:{a},\:{b}\:{solutions}\:{to}\:{the}\:{above}\:{sum}\:{resulting}\:{in}\:{k}\:\in\:\mathbb{Z}^{\ast} \\ $$$$ \\ $$$${if}\:{b}\:=\:{na},\:{n}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{\mathrm{13}{a}\:+\:\mathrm{19}{na}}{{na}^{\mathrm{2}} }\:=\:\frac{{a}\left(\mathrm{13}\:+\:\mathrm{19}{n}\right)}{{na}^{\mathrm{2}} }\:=\:\frac{\mathrm{13}\:+\:\mathrm{19}{n}}{{na}}\:=\:\frac{\mathrm{19}}{{a}}\:+\:\frac{\mathrm{13}}{{na}} \\ $$$$ \\ $$$${gcd}\:=\:\mid{ab}\mid;\:{a},\:{b}\:{solutions}\:{to}\:{the}\:{above}\:{sum}\:{resulting}\:{in}\:{k}\:\in\:\mathbb{Z}^{\ast} \\ $$$$ \\ $$$${if}\:{a}\:\neq\:{mb},\:{a}\:=\:{mb}\:+\:{r};\:{mb},\:{r}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{\mathrm{13}\left({mb}\:+\:{r}\right)\:+\:\mathrm{19}{b}}{\left({mb}\:+\:{r}\right){b}} \\ $$$$\frac{\mathrm{13}}{{b}}\:+\:\frac{\mathrm{19}}{{mb}\:+\:{r}} \\ $$$$ \\ $$$${if}\:{b}\:\neq\:{na},\:{b}\:=\:{na}\:+\:{r};\:{na},\:{r}\:\in\:\mathbb{Z}^{\ast} : \\ $$$$ \\ $$$$\frac{\mathrm{13}{a}\:+\:\mathrm{19}\left({na}\:+\:{r}\right)}{{a}\left({na}\:+\:{r}\right)} \\ $$$$\frac{\mathrm{19}}{{a}}\:+\:\frac{\mathrm{13}}{{na}\:+\:{r}} \\ $$$$ \\ $$$${all}\:{possible}\:{gcds}\:{when}\:{ab}\:{and}\:\mathrm{13}{a}\:+\:\mathrm{19}{b}\:{are}\:{multiple}\:{of}\:{each}\:{other}: \\ $$$$ \\ $$$${if}\:{ab}\:\geq\:\mathrm{13}{a}\:+\:\mathrm{19}{b}:\:{k}\:=\:\frac{{mb}}{\mathrm{13}{m}\:+\:\mathrm{19}}\:;\:\frac{{na}}{\mathrm{19}{n}\:−\:\mathrm{13}}\:\:{k}\:=\:\frac{{a}}{\mathrm{19}}\:+\:\frac{{na}}{\mathrm{13}}\:;\:\:{k}\:=\:\frac{\left({mb}\:+\:{x}\right){b}}{\mathrm{13}\left({mb}\:+\:{x}\right)\:+\:\mathrm{19}{b}}\:;\:{k}\:=\:\frac{{a}\left({na}\:+\:{x}\right)}{\mathrm{13}{a}\:+\:\mathrm{19}\left({na}\:+\:{x}\right)},\:{with}\:\left({a},\:{b}\:\in\:\mathbb{Z}^{\ast} \right),\:\left({m},\:{n}\:\in\:\mathrm{Q}^{\ast} \right)\:\mathscr{H}\underline{\mathscr{H}}\cancel{\underbrace{ }} \\ $$

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