Imagine a number system where the only numbers are those that are 1 greater than a multiple of 20, for instance, 21, 81, 1741. You cannot add or subtract using this number system without the result being a number outside the number system, however it is possible to multiply, as multiplying together two numbers that are each 1 greater than a multiple of 20 will result in a third number that is also 1 greater than a multiple of 20. For example, 21 x 61 = 1281.

‘Prime’ numbers exist in this system, defined as numbers that cannot be formed by multiplying together two smaller numbers in the number system. All actual primes, such as 41, are obviously still prime in this system, but other numbers, such as 21 or 81, are not prime usually, but are in this system.

One well known fact about ordinary numbers is that they are the product of prime numbers in exactly one way, for example 72 = 2x2x2x3x3. However, it is possible for numbers in this special number system to be the product of ‘prime’ (within the system) numbers in more than one way.

What is the smallest number in this number system that is the product of ‘primes’ in two distinct ways?