Hint: To find the greatest possible rate at which the man can walk the given distances in exact number of days take the H.C.F of the three distances 68 km, 102 km and 51 km and consider the unit of the obtained H.C.F as km/day. To find the H.C.F take the product of all the common factors of the three numbers using the prime factorization method.
Complete step by step solution: Here we have been asked to find the greatest possible rate at which a man can walk 68 km, 102 km and 51 km in an exact number of days. That means we need to find the speed of the man. The required greatest possible rate with which the man can walk will be the H.C.F of the three numbers 68, 102 and 51 with the unit of the H.C.F as km/day. So, let us find the H.C.F of the three distances using the prime factorization method. Writing each number as the product of their prime factors we get, $\Rightarrow 68=2\times 2\times 17$ $\Rightarrow 102=2\times 3\times 17$ $\Rightarrow 51=3\times 17$ Now, the H.C.F will be a product of all the common prime factors of the three numbers. Clearly we can see that the only factor common in all of them is 17 so the H.C.F of 68, 102 and 51 is 17. Hence, the rate at which the man should walk is 17 km/day.
Note: Note that you do not have to take the L.C.M of the three numbers. If you are asked to calculate the rate in hours then simply divide the obtained H.C.F 17 by 24 because we know that 1 day consists of 24 hours. Similarly, you can use the different time or distance relations to find the speed in different units of distance and time.