Q-26-58-k-29-0-and-k-N-find-Min-k-

Question Number 173411 by mnjuly1970 last updated on 11/Jul/22

Q :

{(26!)}^{58} + k \overset{29}{\equiv} 0

and, k \in N, find :

Min (k) = ?

Answered by Rasheed.Sindhi last updated on 11/Jul/22

(29 - 1)! \equiv - 1 (m o d 29) [{W i l s o n}^{'} s t h e o r e m]

28! \equiv - 1 + 29 (m o d 29)

28! \equiv 28 (m o d 29)

27! \equiv 1 (m o d 29)

27! \equiv 1 + 29 \times 13 (m o d 29)

27! \equiv 378 (m o d 29)

26! \equiv 14 (m o d 29)

{(26!)}^{2} \equiv 14^{2} \equiv 22 (m o d 29) \dots . . (i)

{(26!)}^{28} \equiv 14^{28} \equiv 1 (m o d 29)

{{(26!)}^{28}}^{2} \equiv 1^{2} (m o d 29) \dots \dots . . (i i)

(i) \times (i i)

{(26!)}^{56} \cdot {(26!)}^{2} \equiv 1 \cdot 22 (m o d 29)

{(26!)}^{58} \equiv 22 - 29 = - 7 (m o d 29)

{(26!)}^{58} + 7 \equiv 0 (m o d 29)

{(26!)}^{58} + 7 + 29 m \equiv 0 (m o d 29)

k = 7 + 29 m [k \in N \Rightarrow m ⩾ 0]

m = 0 : m i n (k) = 7

Commented by mnjuly1970 last updated on 11/Jul/22

thanks alot sir