Solution: Because the remainder of A is 6, it means that the single digit of A is not less than 7, which may be 7, 8, 9; If the company number is 7 and the age is 55, it does not meet the conditions. If the company number is 8 and the age is 62, it does not meet the conditions. If unit number is 9, age is 69. Obviously, the condition is met, that is, A's age is 69.

Because the remainder of B is 9, it means that the sum of the tenth digit and the single digit of B's age is greater than 9. If the sum is 10 and the age is 49, it does not meet the conditions. Similarly, the sum is 1 1, 13, 14, 16.

When B is 57 years old, the difference between them is 12 years old, and when B is 69 years old, they are the same age.