If-7-n-10-7-19-then-find-the-1st-digit-of-the-numer-8-n-4-

Question Number 182718 by mnjuly1970 last updated on 13/Dec/22

If, 7^{n} \overset{10}{\equiv} 7^{19}

t h e n f i n d t h e 1 s t d i g i t

o f t h e n u m e r, 8^{n + 4} .

Commented by JDamian last updated on 13/Dec/22

"1st digit" is the right-most one?

Commented by mnjuly1970 last updated on 13/Dec/22

y e s s i r

e x a m p l e \dots 1234 \overset{10}{\equiv} 4

Commented by Rasheed.Sindhi last updated on 13/Dec/22

{W h a t}^{'} s r e l a t i o n b e t w e e n m & n ?

I s m = n ?

Commented by mnjuly1970 last updated on 13/Dec/22

y e s

Answered by TheSupreme last updated on 13/Dec/22

n = 19

a_{n} = {a : 8^{n} \equiv_{10} a}

8^{1} =_{10} 8

8^{2} =_{10} 4

8^{3} =_{10} 2

8^{4} =_{10} 6

8^{5} =_{10} 8

8^{5 k + i} = {8, 4, 2, 6, 8}

8^{19 + 4} =_{10} 2

Answered by Rasheed.Sindhi last updated on 14/Dec/22

\underline{MOD 10}

7^{2} = 49 \Rightarrow 7^{2} \equiv 9

7^{2} . 7 \equiv 9.7 = 63 \Rightarrow \begin{matrix} 7^{3} \equiv 3 \end{matrix} \dots i

7^{3} . 7 \equiv 3.7 = 21 \Rightarrow 7^{4} \equiv 1

{(7^{4})}^{k} \equiv 1^{k} \Rightarrow \begin{matrix} 7^{4 k} \equiv 1 \end{matrix} \dots . i i

i \times i i : \begin{matrix} 7^{4 k + 3} \equiv 3 \end{matrix}

7^{n} \equiv 7^{19} = 7^{4 (4) + 3} \equiv 3 \equiv 7^{4 k + 3}

n = 4 k + 3

\underline{8^{n + 4} = 8^{4 k + 3 + 4} = 8^{4 k + 7} = ?}

8^{2} \equiv 4

8^{3} \equiv 2 \dots . i i i

8^{4} \equiv 6 \dots . i v

8^{4 k} \equiv 6^{k} \equiv 6^{★} \dots . v

i i i \times i v : 8^{7} \equiv 2 \dots v i

v \times v i : 8^{4 k + 7} \equiv 2

\begin{matrix} 8^{n + 4} \equiv 2 \end{matrix}

\begin{matrix} ^{★} 6^{k} \equiv 6 (m o d 10) \end{matrix}

Commented by mnjuly1970 last updated on 14/Dec/22

t h x a l o t s i r R a s h e e d