I am a number greater than 50 and less than 60. I am divisible by 2. If you subtract from me one, I become divisi- • ble by 5.

Let's call the number x.

From the given information, we know the following conditions must be satisfied:
1. x > 50 and x < 60
2. x is divisible by 2, which means x is an even number.
3. If you subtract 1 from x, the result is divisible by 5.

Now we can find the solution step by step:

1. x > 50 and x < 60: This narrows down the possible values for x to the range 51, 52, 53, 54, 55, 56, 57, 58, 59.

2. x is divisible by 2: This means x is an even number. From the range of possible values, we can eliminate the odd numbers 51, 53, 55, 57, and 59. The possible values for x are now 52, 54, 56, and 58.

3. If you subtract 1 from x, the result is divisible by 5: Let's test the remaining possible values:
- If x = 52, then subtracting 1 gives 51, which is not divisible by 5. So 52 is not the solution.
- If x = 54, then subtracting 1 gives 53, which is not divisible by 5. So 54 is not the solution.
- If x = 56, then subtracting 1 gives 55, which is divisible by 5. So 56 is a possible solution.
- If x = 58, then subtracting 1 gives 57, which is not divisible by 5. So 58 is not the solution.

Therefore, the number that satisfies all the given conditions is x = 56.

Answer: $\boxed{x = 56}$