The H.C.F. of the numbers, 55125 and 2250, is of the form a² × b^a. What are the respective values of a and b?

The numbers, 55125 and 2250, can be prime factorized as:

55125 = 3 × 3 × 5 × 5 × 5 × 7 × 7

              = 3² × 5³ × 7²

And 2250 = 2 × 3 × 3 × 5 × 5 × 5

                     = 2 × 3² × 5³

∴ H.C.F. of (55125, 2250) = 3² × 5³

Comparing 3² × 5³ with a² × b^a, we obtain

a = 3 and b = 5

Thus, the values of a and b are 3 and 5 respectively.