Find the greatest number which divides 290 and 538 leaving remainders 3 and 5 respectively.

If a number divides 290 leaving a remainder of 3, it must exactly divide $290 - 3$.

$$290 - 3 = 287$$

If a number divides 538 leaving a remainder of 5, it must exactly divide $538 - 5$.

$$538 - 5 = 533$$

$$287 = 7 \times 41$$ $$533 = 13 \times 41$$

The HCF is the product of the common prime factors raised to the lowest power.
The only common prime factor is 41.

$$\text{HCF}(287, 533) = 41$$