# Problem of the Week Problem E A Small Subset

There are $$90\,000$$ five-digit positive integers. However, only some of these five-digit integers satisfy the following conditions:

• the middle digit is $$0$$,

• the ten thousands digit and the tens digit are equal,

• the thousands digit and the ones (units) digit are equal, and

• the number has exactly $$5$$ prime factors. All of these prime factors are odd and none are repeated.

Determine all five-digit positive integers that satisfy the given conditions.