n is an odd positive integer. The number 7ⁿ + 3ⁿ is always divisible by:

When the index is ODD, aⁿ + $b$ⁿ always has $a+b$ as a factor, because aⁿ + $b$ⁿ = $(a+b)$(aⁿ⁻¹ − aⁿ⁻²$b$ + … + $b$ⁿ⁻¹). Here $a+b=7+3=10$. Checking: $n=1$ gives 10, $n=3$ gives $343+27=370=10\times 37$, and $n=5$ gives $16807+243=17050=10\times 1705$. None of these is a multiple of 4, 21 or 49 — multiplying the two bases to get 21 is the trap. The rule really does need $n$ odd: $7^2+3^2=58$ is not a multiple of 10.