Modulo problems

$2222^{5555}$ is even and $5555^{2222}$ is odd, therefore their sum cannot be a multiple of 2.

$2222^{5555}+5555^{2222}\equiv (-1)^{5555}+(-1)^{2222}=-1+1=0$ modulus 3, so it is a multiple of 3.

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With a little help of a computer: $2222^{5555}+5555^{2222}=$ $3\times 7\times11^{2222}\times 23\times 101^{2222}\times 5\,657\times 153\,319\times 340\,169\times 1\,182\,611\times 14\,287\,057\times \prod p_i^{r_i}$ with all primes $p_i>10^9$ .