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Question Number 136830 by bramlexs22 last updated on 26/Mar/21

Determine all solutions in the integers of the following Diophantine equations: (a)56x+72y=40 (b)24x+138y=18 (c)221x+35y=11\n

Answered by floor(10²Eta[1]) last updated on 26/Mar/21

(a)56x+72y=40  ⇒7x+9y=5  7x+9y=5  (look mod 3)  x≡2(mod 3)⇒x=3a+2  7(3k+2)+9y=5  y=−1−(7/3)a⇒3∣a⇒a=3k  x=9k+2, y=−1−7k  (b)24x+138y=18  ⇒4x+23y=3 (look mod 2)  y≡1(mod 2)⇒y=2a+1  4x+23(2a+1)=3  x=−5−((23)/2)a⇒2∣a⇒a=2k  x=−5−23k, y=4k+1  (c)221x+35y=11 (look mod 5)  x≡1(mod 5)⇒x=5a+1  221(5a+1)+35y=11  y=−((221)/7)a−6⇒7∣a⇒a=7k  x=35k+1, y=−221k−6

Extra close brace or missing open brace 7x+9y=5 7x+9y=5(lookmod3) x2(mod3)x=3a+2 7(3k+2)+9y=5 y=173a3aa=3k x=9k+2,y=17k Extra close brace or missing open brace 4x+23y=3(lookmod2) y1(mod2)y=2a+1 4x+23(2a+1)=3 x=5232a2aa=2k x=523k,y=4k+1 Extra close brace or missing open brace x1(mod5)x=5a+1 221(5a+1)+35y=11 y=2217a67aa=7k x=35k+1,y=221k6

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