The number n is the produce of the first 1000 positive integers and can be written as 1000! .we say "1000 factorial."this is, n=1000! =.n is divisible by 5, 25, 125, of these factors is a power of 5.that is 5=5^1, 25=5^2, 125=5^3,625=5^4, and so on. determine the largest power of 5 that divides n.

From 1000 to 1, there are 200 numbers that are divisible by 5, which means that 1000! is divisible by at least 5^200.Then there are 40 numbers that are divisible by 25 (5^2), so this adds another 5^40.Then there are 8 numbers divisible by 125 (5^3), so this adds another 5^8.Finally, 1 number is divisible by 625 (5^4), so this adds 5^1.
Therefore, the total sum is 5^(200+40+8+1) = 5^249. Therefore, the largest N = 249.

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