How many positive integers less than 500 are the product of two distinct (meaning different) primes, each of which is greater than 10?
i don't think there is a snap way to do this list the primes between 10 and 50 and see how many give you products less than 500
ok
A useful clue is that since 23^2 is greater than 500, any product of two primes less than 500, each of which is greater than ten has one of the primes between 11 and 19.
11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 i think this is your list now you can multiply but \(11\times 47\) is too large so only up to 43 you can multiply 11 by the other 9 primes 13 by 7 others because \(13\times 41\) is too big, and so on
So you just have to solve for 11, 13, 17, and 19. 11*(11,13,17,19,23,29,31,37,41, or 43) 13*(13,17,19,23,29,31, or 37) etc.
And avoid counting 11*13 and 13*11 etc. separately.
of course i made a mistake because my method counts \(13\times 11\) twice
but also they have to be distinct, so don't count \(13\times 13\) etc
Oh, I missed that part. Yeah, avoid the squares.
i got 23 choices
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