Q-find-the-largest-value-of-such-that-the-positive-integers-a-b-gt-1-satisfy-a-b-b-a-a-b-b-a-5329-

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

Q . find the largest value of such that the

positive integers a, b > 1 satisfy .

a^{b} . b^{a} + a^{b} + b^{a} = 5329

Commented by MJS_new last updated on 18/Nov/22

the largest value of w h a t ?

anyway if a, b \in Z^{+} ∖ {1} we have these solutions :

a = 3 \land b = 4 \lor a = 4 \land b = 3

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

a^{b} . b^{a} + a^{b} + b^{a} = 5329

a^{b} (b^{a} + 1) + b^{a} + 1 = 5330

(a^{b} + 1) (b^{a} + 1) = 5330 = 2 \cdot 5 \cdot 13 \cdot 41

k ∣ 5330 :

a^{b} + 1 = k \land b^{a} + 1 = 5330 / k

\Rightarrow k = 65, 82

{\begin{cases} a^{b} + 1 = 65 \land b^{a} + 1 = 82 \\ a^{b} + 1 = 82 \land b^{a} + 1 = 65 \end{cases}

{\begin{cases} a^{b} = 64 \land b^{a} = 81 \Rightarrow 4^{3} = 64 \land 3^{4} = 81 \\ a^{b} = 81 \land b^{a} = 64 \Rightarrow 3^{4} = 81 \land 4^{3} = 64 \end{cases}

{a, b} = {3, 4}