Find-all-positive-integers-n-for-which-5-n-1-is-divisible-by-7-

Question Number 113073 by bobhans last updated on 11/Sep/20

Find all positive integers n for

which 5^{n} + 1 is divisible by 7

Answered by john santu last updated on 11/Sep/20

f i n d a l l p o s i t i v e i n t e g e r s n s u c h

t h a t 5^{n} + 1 d i v i s i b l e b y 7

(s o l) : 5^{n} + 1 \equiv 0 (m o d 7)

T h i s c a n b e r e w r i t t e n 5^{n} \equiv - 1 \equiv 125 (m o d 7)

H o w e v e r b y {F e r m a t}^{'} s L i t t l e T h e o r e m

5^{6} \equiv 1 (m o d 7) . M o r e v e r i t e a s y t o

c h e c k t h a t 5^{2} \neq 1 (m o d 7) a n d

5^{3} \neq 1 (m o d 7) . H e n c e w e m u s t

h a v e n \equiv 3 (m o d 6) o r n = 3 + 6 k, k \in Z^{+}

\frac{J S}{a m a t h f a r m e r}

Commented by bemath last updated on 11/Sep/20

✓ thank you

Facebook Tweet Pin