2007 SMT Algebra Round Problem 6

Problem

What is the largest prime factor of $4^9+9^4$ ?

Solution

By just bashing it out, we get $4^9=2^{18}=2^{10}\times2^8=1024\times256=262144$ and $9^4=9^2\times9^2=81\times81=6561$ . After adding these two up, we get $4^9+9^4=262144+6561=268705$ . By just looking at this, we see that it is divisible by $5$ , so let's divide by $5$ to get $53741$ . After trying all the primes from $7$ to $61$ ( $5$ won't work because this number does not end in $5$ or $0$ ), we get that this number is divisible by $61$ , and our quotient is $881$ . Because no prime less than $61$ divided $53741$ , they won't divide $881$ either, and because $61^2 > 881$ , $881$ must be prime, so our primes are $5, 61,$ and $881$ . Because $881$ is the greatest out of all of these, our answer is $\boxed{\mathrm{881}}$ .

~Yuhao2012

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2007 SMT Algebra Round Problem 6

Problem

Solution