Brainsteem: A Prime STEEMIT Number Puzzle

I just made this number puzzle this afternoon to kick off my writing on Steemit. I hope you find it amusing.

Take the word STEEMIT and replace each letter with a distinct numeral from 0 to 9 to create a 7-digit number. Each repeating letter must be replaced by the same numeral and the letter S cannot be zero.

What is the smallest such number that is a prime number?

For example, if STEEMIT --> 3577245 then this is not a prime number, as divisible by 5.

Bonus upvote if you include a method with your solution. There is a logic to solve this more quickly than random!

Brainsteem Pro: I think the above is quite doable with pen, paper and a prime number checker, but for those of you who enjoy a computational challenge, try finding every possible prime number.

I have created many mathematical problems and will slowly be adding them to Steemit. They will range in difficulty from primary competitions and national challenges to pre-Olympiad questions and some computational mathematics. There will also be other kinds of puzzles, such as on logic or word puzzles, plus reviews of any clever online games; pretty much anything I find amusing. This also gives me the opportunity to freshen them up and tweak them a little.