1. **Problem statement:** Find the greatest number which divides 70 and 125 leaving remainders 5 and 8 respectively. 2. **Understanding the problem:** If a number $d$ divides 70 leaving remainder 5, then $70 - 5 = 65$ is divisible by $d$. Similarly, if $d$ divides 125 leaving remainder 8, then $125 - 8 = 117$ is divisible by $d$. 3. **Formula and approach:** The greatest number $d$ that divides both 65 and 117 exactly is the greatest common divisor (GCD) of 65 and 117. 4. **Calculate GCD:** - Prime factorization of 65: $65 = 5 \times 13$ - Prime factorization of 117: $117 = 3^2 \times 13$ 5. **Find common factors:** Both have 13 in common. 6. **Therefore,** the greatest number $d = 13$. **Final answer:** $13$