montrer que 7 divise 2222^(5555) +5555^(2222)


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Question Number 158794 by SANOGO last updated on 08/Nov/21

$m o n t r e r q u e 7 d i v i s e$ $2222^{5555} + 5555^{2222}$
	Answered by Rasheed.Sindhi last updated on 09/Nov/21

	${W e}^{'} l l d e t e r m i n e r e m a i n d e r s$ $o f 2222^{5555} & 5555^{2222} s e p a r a t e l y$ $a n d t h e n a d d t h e m :$ $▸ 2222^{5555} \equiv x (m o d 7)$ $2222 \equiv 3 (m o d 7)$ $2222^{5} \equiv 3^{5} \equiv 5 (m o d 7) . . . . . . . . . . (i)$ $∵ \gcd (2222, 7) = 1 & ϕ (7) = 6$ $∴ 2222^{6} \equiv 1 (m o d 7)$ ${(2222^{6})}^{925} \equiv {(1)}^{925} (m o d 7)$ $2222^{5550} \equiv 1 (m o d 7) . . . . . . . . . . . . (i i)$ $(i) \times (i i) :$ $2222^{5} . 2222^{5550} \equiv 5.1 (m o d 7)$ $2222^{5555} \equiv 5 (m o d 7) . . . . . . . . . . . . . A$ $▸ 5555^{2222} \equiv y (m o d 7)$ $\gcd (5555, 7) = 1 & ϕ (7) = 6$ $5555^{6} \equiv 1 (m o d 7)$ ${(5555^{6})}^{370} \equiv {(1)}^{370} (m o d 7)$ $5555^{2220} \equiv 1 (m o d 7) . . . . . . . . (i i i)$ $5555 \equiv 4 (m o d 7)$ $5555^{2} \equiv 4^{2} \equiv 2 (m o d 7) . . . . . . . . (i v)$ $(i i i) \times (i v) :$ $5555^{2222} \equiv 2 (m o d 7) . . . . . . . . . . . . . . . . . B$ $A + B :$ $2222^{5555} + 5555^{2222} \equiv 5 + 2 \equiv 0 (m o d 7)$ $H e n c e 7 d i v i d e s g i v e n e x p r e s s i o n .$ $\overset{≼ \underset{}{\overset{}{FORMULA USED ≽}}}{{\begin{matrix} \underset{p r o v i d e d t h a t \gcd (a, m) = 1}{a^{ϕ (m)} \equiv 1 (m o d m)} \end{matrix}}^{}}$
	Commented by SANOGO last updated on 09/Nov/21

	$m e r c i b i e n$
	Answered by Rasheed.Sindhi last updated on 09/Nov/21

	$MOD 7$ $2222^{5555} + 5555^{2222}$ $= 2222^{6 \times 925 + 5} + 5555^{6 \times 370 + 2} [∵ ϕ (7) = 6]$ $= {(2222^{6})}^{925} . 2222^{5} + {(5555^{6})}^{370} . 5555^{2}$ $= (1^{925}) (2222^{5} + (1^{370}) (5555^{2})$ $[∵ 2222^{6} \equiv 1 & 5555^{6} \equiv 1 (m o d 7)]$ $= 2222^{5} + 5555^{2}$ $= 3^{5} + 4^{2} [∵ 2222 \equiv 3 & 5555 \equiv 4 (m o d 7)]$ $= 5 + 2 = 7 = 0$ $H e n c e$ $7 ∣ (2222^{5555} + 5555^{2222})$
	Commented by SANOGO last updated on 10/Nov/21

	$m e r c i b i e n$
	Commented by SANOGO last updated on 10/Nov/21

	$m e r c i b i e n$
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