Two whole numbers, each greater than 1, are co-prime and their LCM is 143. What is their sum?

For co-prime numbers the HCF is 1, so LCM × HCF = product tells us the product is also 143. Now $143=11\times 13$, and since both numbers must exceed 1 the only choice is 11 and 13 (which are indeed co-prime). Their sum is 24. Choosing 22 or 26 pairs a number with itself, which would destroy co-primality, and 144 is 143 + 1.