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# Number Theory

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208
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N = 1991*1993*1995*1997*1999*2001*2003.

What is the sum of the hundreds, tens and units digits of N?

Jul 4, 2021

#1
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Since we need to compute the last 3 digits, we can take the whole expression mod 1000:

$$1991\cdot1993\cdot1995\cdot1997\cdot1999\cdot2001\cdot2003 (\text{mod 1000})\\ =-9\cdot-7\cdot-5\cdot-3\cdot-1\cdot1\cdot3 (\text{mod 1000})\\ =-2835 (\text{mod 1000})\\ =165(\text{mod 1000})$$

The sum of the hundreds, tens, and units digit is $$1+6+5=\boxed{12}$$

Jul 4, 2021