Step 1: If the required number divides 764 leaving remainder 8, it exactly divides $764-8=756$.
Step 2: Similarly it exactly divides $1198-10=1188$.
Step 3: So the required number $=$ HCF(756, 1188).
Step 4: Prime factorisation:
$$756=2^2\times 3^3\times 7$$
$$1188=2^2\times 3^3\times 11$$
Step 5: HCF $=2^2\times 3^3=4\times 27=108$.
Step 6: Check: $756\div 108=7$ ✓ and $1188\div 108=11$ ✓
Answer: $108$