determiner-le-reste-de-la-division-euclidienne-de-10-100-par-105-

Question Number 162118 by SANOGO last updated on 27/Dec/21

d e t e r m i n e r l e r e s t e d e l a d i v i s i o n e u c l i d i e n n e d e

10^{100} p a r 105

Commented by SANOGO last updated on 27/Dec/21

t h a n k y o u

Commented by SANOGO last updated on 27/Dec/21

m e r c i b i e n

Commented by mr W last updated on 27/Dec/21

i n f o l l o w i n g s t u p i d m e t h o d

w i t h “ {\overset{R}{=}}^{″} i m e a n

“ h a s t h e s a m e r e m a i n d e r {a s}^{″} .

10^{100} = 10 \times 10^{99} = 10 \times {(1000)}^{33}

= 10 \times {(9 \times 105 + 55)}^{33}

\overset{R}{=} 10 \times 55^{33} = 10 \times 55 \times {(3025)}^{16}

= 10 \times 55 \times {(28 \times 105 + 85)}^{16}

\overset{R}{=} 10 \times 55 \times 85^{16} = 10 \times 55 \times {(68 \times 105 + 85)}^{8}

\overset{R}{=} 10 \times 55 \times 85^{8}

\overset{R}{=} 10 \times 55 \times 85^{4}

\overset{R}{=} 10 \times 55 \times 85^{2}

\overset{R}{=} 10 \times 55 \times 85

\overset{R}{=} 25

i . e . t h e r e m a i n d e r i s 25 w h e n 10^{100}

i s d i v i d e d b y 105 .

Commented by Rasheed.Sindhi last updated on 27/Dec/21

\lor \cap i \subset\in sir!

I f y o u r^{'} {s t u p i d}^{'} m e t h o d i s s o w o n d e r f u l

w h a t y o u r^{'} {w o n d e r f u l}^{'} m e t h o d w i l l b e!!!

Commented by Rasheed.Sindhi last updated on 27/Dec/21

T h e f o l l o w i n g m e t h o d a l s o w o r k s w e l l :

\underset{\underset{\underset{o n d i v i d i n g 6}{s a m e r e m a i n d e r}}{x & y h a v e}}{x \equiv y (m o d 6)} \Rightarrow \underset{\underset{\underset{o n d i v i d i n g 105}{s a m e r e m a i n d e r}}{10^{x} & 10^{y} h a v e}}{10^{x} \equiv 10^{y} (m o d 105)}

\Rightarrow 10^{6 x + k} \equiv 10^{k}

10^{100} = 10^{6 \times 16 + 4} \equiv 10^{4} \equiv 25 (m o d 105)

Commented by mr W last updated on 27/Dec/21

t h i s i s a s m a r t m e t h o d!

Commented by Rasheed.Sindhi last updated on 27/Dec/21

T han k s sir!