The greatest number that divides 245 and 1029 leaving remainder 5 in each case is:

Subtract the remainder first: 245 − 5 = 240 and 1029 − 5 = 1024. The required number is HCF(240, 1024). Since $240=2^4\times 3\times 5$ and $1024=2^{10}$, the HCF is $2^4=16$ (and 16 > 5, so a remainder of 5 is possible).