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Question Number 104413 by Anindita last updated on 21/Jul/20

$$\mathbf{The}\mathbf{sum}\mathbf{of}\mathbf{two}\mathbf{integers}\mathbf{is}304$$$$\mathbf{and}\mathbf{their}\mathbf{GCD}\mathbf{is}19.\mathbf{What}\mathbf{are}\mathbf{the}$$$$\mathbf{numbers}?$$

Answered by Rasheed.Sindhi last updated on 22/Jul/20

$$Integers:19x,19y$$$$Sum:19x+19y=304$$$$x+y=16$$$$y=16-x$$$$Requiredintegersare19x,19(16-x)$$$$\mathrm{\forall }x\in \mathbb{Z},with\mathrm{GCD}(x,16-x)=1$$

Commented by floor(10²Eta[1]) last updated on 21/Jul/20

$$\mathrm{you}\mathrm{are}\mathrm{including}\mathrm{all}\mathrm{the}\mathrm{cases}\mathrm{when}$$$$\gcd (\mathrm{x},\mathrm{y})\ne 1\mathrm{which}\mathrm{is}\mathrm{not}\mathrm{correct}$$

Commented by Rasheed.Sindhi last updated on 22/Jul/20

$$YesthisrestrictionIforgot.$$$$\mathcal{T}hankyou!$$

Answered by 1549442205PVT last updated on 21/Jul/20

$$\mathrm{Since}\mathrm{GCD}(\mathrm{a},\mathrm{b})=19\Rightarrow \{\begin{array}{l}\mathrm{a}=19\mathrm{m}\\ \mathrm{b}=19\mathrm{n}\end{array}$$$$\mathrm{where}\gcd (\mathrm{m},\mathrm{n})=1$$$$\mathrm{From}\mathrm{the}\mathrm{hypothesis}\mathrm{a}+\mathrm{b}=304\mathrm{we}\mathrm{get}$$$$19(\mathrm{m}+\mathrm{n})=304\Rightarrow \mathrm{m}+\mathrm{n}=16.$$$$\mathrm{WLOG}\mathrm{we}\mathrm{suppose}\mathrm{m}<\mathrm{n}.\mathrm{Then}\mathrm{we}\mathrm{have}$$$$\mathrm{the}\mathrm{following}\mathrm{cases}:$$$$(\mathrm{note}\mathrm{that}\gcd (\mathrm{m},\mathrm{n})=1\Rightarrow \mathrm{m}\mathrm{is}\mathrm{odd})$$$$\mathrm{i})\mathrm{m}=1\Rightarrow \mathrm{n}=15\Rightarrow (\mathrm{a},\mathrm{b})=(19,285)$$$$\mathrm{ii})\mathrm{m}=3\Rightarrow \mathrm{n}=13\Rightarrow (\mathrm{a},\mathrm{b})=(57,247)$$$$\mathrm{iii})\mathrm{m}=5\Rightarrow \mathrm{n}=11\Rightarrow (\mathrm{a},\mathrm{b})=(209,95)$$$$\mathrm{iv})\mathrm{m}=7\Rightarrow \mathrm{n}=9\Rightarrow (\mathrm{a},\mathrm{b})=(153,151)$$$$\mathrm{Thus}\mathrm{all}\mathrm{the}\mathrm{pairs}(\mathrm{a},\mathrm{b})\mathrm{of}\mathrm{positive}$$$$\mathrm{integers}\mathrm{which}\mathrm{we}\mathrm{need}\mathrm{to}\mathrm{find}\mathrm{as}:$$$$(\mathrm{a},\mathrm{b})\in \{(19;285),(57,247),(209;95),(153;151)\}$$$$\mathbf{and}\mathbf{the}\mathbf{permutations}\mathbf{of}\mathbf{them}$$$$$$

Commented by floor(10²Eta[1]) last updated on 21/Jul/20

$$\gcd (153,151)=1\ne 19.$$$$\mathrm{the}\mathrm{last}\mathrm{case}\mathrm{should}\mathrm{be}$$$$(\mathrm{a},\mathrm{b})=(133,171)$$

Commented by floor(10²Eta[1]) last updated on 21/Jul/20

$$\mathrm{m}=2\mathrm{s}+1,\mathrm{n}=2\mathrm{t}+1$$$$\mathrm{m}+\mathrm{n}=2\mathrm{s}+2\mathrm{t}+2=16\Rightarrow \mathrm{s}+\mathrm{t}=7$$$$\mathrm{s}=\mathrm{k},\mathrm{t}=7-\mathrm{k}$$$$\mathrm{m}=2\mathrm{k}+1,\mathrm{n}=15-2\mathrm{k}$$$$2\mathrm{k}+1<15-2\mathrm{k}\Rightarrow \mathrm{k}⩽3$$$$\mathrm{so}\mathrm{all}\mathrm{the}\mathrm{pairs}(\mathrm{a},\mathrm{b})\mathrm{of}\mathrm{integers}\mathrm{are}:$$$$(\mathrm{a},\mathrm{b})=\{19(2\mathrm{k}+1),19(15-2\mathrm{k})\}\mathrm{\forall }\mathrm{k}⩽3$$$$$$

Commented by 1549442205PVT last updated on 22/Jul/20

$$\mathrm{Thank}\mathrm{you}\mathrm{sir},\mathrm{i}\mathrm{mistaked}\mathrm{this}\mathrm{case}$$

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