The largest number that divides 245 and 1029, leaving remainders 5 and 5 respectively, is

The number divides $245-5 = 240$ and $1029-5 = 1024$. Required number $= \gcd(240, 1024)$. $240 = 2^4\cdot3\cdot5$, $1024 = 2^{10}$, so $\gcd = 2^4 = 16$.