elementary-mathematics-if-13-9-51-k-1-k-N-then-k-min-

Question Number 126765 by mnjuly1970 last updated on 24/Dec/20

\dots e l e m e n t a r y m a t h e m a t i c s \dots

i f 13 ∣ 9^{51} + k + 1, k \in N

t h e n k_{(m i n)} = ?

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

9^{51} + k + 1 \equiv 0 (\mod 13)

9^{51} = {(9^{4})}^{12 + 3} \equiv 9^{3} \equiv {(- 4)}^{3} = - 64 \equiv 1 (\mod 13)

9^{51} + k + 1 \equiv k + 2 \equiv 0 (\mod 13)

\Rightarrow k \equiv 11 (\mod 13)

k_{\min} = 11

Commented by mnjuly1970 last updated on 24/Dec/20

g r a t e f u l . .

Commented by JDamian last updated on 25/Dec/20

{(9^{4})}^{12 + 3} = {(9^{4})}^{15} = 9^{60} \neq 9^{51}

9^{51} = {(9^{4})}^{12} \times 9^{3} \neq {(9^{4})}^{12 + 3}

Commented by floor(10²Eta[1]) last updated on 25/Dec/20

you multiply 4.12 first bro {that}^{'} s basic

{(9^{4})}^{12 + 3} \neq {(9^{4})}^{(12 + 3)}