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Question Number 123304 by mordo last updated on 24/Nov/20

$$gcd(a;b)+lcm(a;b)=111$$$$findaandb$$

Commented by mr W last updated on 24/Nov/20

$$a=1$$$$b=110$$

Answered by MJS_new last updated on 24/Nov/20

$$\left(1\right)\gcd (a,b)=1\Rightarrow \mathrm{lcm}(a,b)=ab=110$$$$\left(2\right)\gcd (a,b)=k\ne 1\Rightarrow a=kx\wedge b=ky\Rightarrow$$$$\Rightarrow \mathrm{lcm}(a,b)=kxy\Rightarrow k+kxy=111\iff$$$$\iff k=\frac{111}{xy+1}\mathrm{with}\gcd (x,y)=1$$$$\mathrm{in}\mathrm{both}\mathrm{cases}\mathrm{we}\mathrm{can}\mathrm{try}\mathrm{to}\mathrm{find}\mathrm{solutions}$$$$\mathrm{for}a<b\mathrm{I}\mathrm{get}\mathrm{these}\mathrm{pairs}:$$$$ab$$$$1110$$$$255$$$$3108$$$$522$$$$1011$$$$1227$$$$3774$$

Answered by floor(10²Eta[1]) last updated on 24/Nov/20

$$\mathrm{let}\mathrm{d}=\gcd (\mathrm{a},\mathrm{b})\Rightarrow \mathrm{a}=\mathrm{dx},\mathrm{b}=\mathrm{dy}$$$$\mathrm{where}\gcd (\mathrm{x},\mathrm{y})=1$$$$\mathrm{d}+\mathrm{dxy}=111\therefore \mathrm{d}(1+\mathrm{xy})=111$$$$\mathrm{d}\mid 111\Rightarrow \mathrm{d}=1,3,37,111$$$$\mathrm{Case}1:\mathrm{d}=1\Rightarrow \mathrm{xy}=\mathrm{ab}=110=11.5.2$$$$\mathrm{a}=11.5.2,\mathrm{b}=1$$$$\mathrm{a}=11.5,\mathrm{b}=2$$$$\mathrm{a}=11.2,\mathrm{b}=5$$$$\mathrm{a}=11,\mathrm{b}=2.5$$$$\mathrm{Case}2:\mathrm{d}=3\Rightarrow \mathrm{xy}=36=2^2.3^2$$$$\mathrm{x}=2^2.3^2,\mathrm{y}=1\Rightarrow \mathrm{a}=2^2.3^3,\mathrm{b}=3$$$$\mathrm{x}=3^2,\mathrm{y}=2^2\Rightarrow \mathrm{a}=3^3,\mathrm{b}=2^2.3$$$$\mathrm{Case}3:\mathrm{d}=37\Rightarrow \mathrm{xy}=2$$$$\mathrm{x}=2,\mathrm{y}=1\Rightarrow \mathrm{a}=37.2,\mathrm{b}=37$$$$\mathrm{Case}4:\mathrm{d}=111(\mathrm{no}\mathrm{sol}.)$$$$(\mathrm{a},\mathrm{b})=\{(110,1),(55,2),(22,5),(11,10)$$$$(108,3),(27,12),(74,37)\}$$$$\mathrm{and}\mathrm{the}(\mathrm{b},\mathrm{a})\mathrm{pairs}$$

Commented by mordo last updated on 25/Nov/20

$$thanks$$

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