Find out the mistake in Modulo Arithmetic

Please help me to find out an error.

Case 1 : $2222^{5555}=3^{5555}(mod7) = 243^{1111}(mod 7) = 5^{1111}(mod 7) = 5.5^{1110}(mod7)$ $= 5.625^{370}(mod 7) = 5.2^{370}(mod 7)= 5.2.2^{369}(mod 7)$ $= 10.8^{123}(mod 7) = 10.1^{123}(mod 7) = 10(mod 7) =3$

Case 2: $2222^{5555} = 3^{5555}(mod 7)= 3^{5}.3^{5550}(mod 7)= 3^{5}.729^{925}(mod 7)$ $= 3^{5}.1^{925}(mod7) = 3^{5}(mod 7) = 5$

In the two cases we got different answer. Which is the wrong one? And where is the mistake? Please help me as I am very weak in this area. Thanks

#HelpMe! #MathProblem

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`2 \times 3`	$2 \times 3$
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Comments

Case 1 is wrong. Your mistake is to identify $5\times5^{1110}$ with $5\times625^{370}$ modulo $7$ . Presumably you are trying to replace $5$ by $625=5^4$ , but $370 \neq 1100 \div 4$ .

Case 2 is right.

A much quicker way of getting there is to note that $3^6 = 1$ modulo $7$ , and so $2222^{5555} \; = \; 3^{5555} \; = \; 3^5 \; = \; 243 \; = \; 5$ modulo $7$ . We are using one case of a general result which states that $x^{p-1} \,=\, 1$ modulo $p$ for any prime $p$ and any integer $x$ which is coprime to $p$ . This tells us that $3^{5550} = 1$ modulo $7$ .

Mark Hennings - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$