Remainder problems:

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Blazing goIITian

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These are quite easy if you know congruency...

7¹⁰³ is divided by 5

From Fermat Little Theorem,

$7^4equiv 1mod5Rightarrow 7^{103}equiv343equiv3mod5$

32^{32power 32} is divided by 7

$32^6equiv 1mod7Rightarrow 32^{32^{32}}equiv2^{32}equiv2^{30}cdot2^2equiv4mod7$

So leaves remainder 4 when divided by 7..

32³² is divided by 3

$32^2\equiv1\mod3\Rightarrow 32^{32}\equiv1\mod3$

2222⁵⁵⁵⁵ + 5555²²²² is divided by 7

$2222^6\equiv1\mod7\\\Rightarrow2222^{6\times925}\equiv1\mod7\\\Rightarrow 2222^{5555}\equiv2222^5\equiv3^5\equiv5\mod7$

$5555^6\equiv1\mod7\\\Rightarrow5555^{6\times370}\equiv1\mod7\\\Rightarrow 5555^{2222}\equiv5555^2\equiv4^2\equiv2\mod7$

So, 2222⁵⁵⁵⁵ + 5555²²²² leaves remainder 0 when divided by 7

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