Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 180877 by Sheshdevsahu last updated on 18/Nov/22

$$\mathbf{Q}.\mathbf{find}\mathbf{the}\mathbf{largest}\mathbf{value}\mathbf{of}\mathbf{such}\mathbf{that}\mathbf{the}$$ $$\mathbf{positive}\mathbf{integers}\mathbf{a},\mathbf{b}>1\mathbf{satisfy}.$$ $${\mathbf{a}}^{\mathbf{b}}.{\mathbf{b}}^{\mathbf{a}}+{\mathbf{a}}^{\mathbf{b}}+{\mathbf{b}}^{\mathbf{a}}=5329$$

Commented byMJS_new last updated on 18/Nov/22

$$\mathrm{the}\mathrm{largest}\mathrm{value}\mathrm{of}what?$$ $$\mathrm{anyway}\mathrm{if}a,b\in {\mathbb{Z}}^{+}\mathrm{\setminus }\left\{1\right\}\mathrm{we}\mathrm{have}\mathrm{these}\mathrm{solutions}:$$ $$a=3\wedge b=4\vee a=4\wedge b=3$$

Answered by Rasheed.Sindhi last updated on 19/Nov/22

$${\mathbf{a}}^{\mathbf{b}}.{\mathbf{b}}^{\mathbf{a}}+{\mathbf{a}}^{\mathbf{b}}+{\mathbf{b}}^{\mathbf{a}}=5329$$ $${\mathrm{a}}^{\mathrm{b}}({\mathrm{b}}^{\mathrm{a}}+1)+{\mathrm{b}}^{\mathrm{a}}+1=5330$$ $$({\mathrm{a}}^{\mathrm{b}}+1)({\mathrm{b}}^{\mathrm{a}}+1)=5330=2\cdot 5\cdot 13\cdot 41$$ $$k\mid 5330:$$ $${\mathrm{a}}^{\mathrm{b}}+1=k\wedge {\mathrm{b}}^{\mathrm{a}}+1=5330/k$$ $$\Rightarrow k=65,82$$ $$\{\begin{array}{l}{\mathrm{a}}^{\mathrm{b}}+1=65\wedge {\mathrm{b}}^{\mathrm{a}}+1=82\\ {\mathrm{a}}^{\mathrm{b}}+1=82\wedge {\mathrm{b}}^{\mathrm{a}}+1=65\end{array}$$ $$\{\begin{array}{l}{\mathrm{a}}^{\mathrm{b}}=64\wedge {\mathrm{b}}^{\mathrm{a}}=81\Rightarrow 4^3=64\wedge 3^4=81\\ {\mathrm{a}}^{\mathrm{b}}=81\wedge {\mathrm{b}}^{\mathrm{a}}=64\Rightarrow 3^4=81\wedge 4^3=64\end{array}$$ $$\{\mathrm{a},\mathrm{b}\}=\{3,4\}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com