The discussion focuses on calculating the remainder of the expression (2^1000) divided by 7. Participants suggest starting with smaller powers of 2 to identify a repeating pattern in the remainders. The established pattern reveals that the remainders cycle every three terms: 1, 2, 4. Consequently, since 1000 modulo 3 equals 1, the remainder of (2^1000) divided by 7 is 2.
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HallsofIvy said:Since that is a rather large power of two, I suggest that you start with simple examples and see if you can find a pattern:
[itex]2^0= 1[/itex]. Remainder on division by 7: 1
[itex]2^1= 2[/itex]. Remainder on division by 7: 2
[itex]2^2= 4[/itex]. Remainder on division by 7: 4
[itex]2^3= 8[/itex]. Remainder on division by 7: 1
[itex]2^4= 16[/itex]. Remainder on division by 7: 2
[itex]2^5= 32[/itex]. Remainder on division by 7: 4
[itex]2^6= 64[/itex]. Remainder on division by 7: 1
[itex]2^7= 128[/itex]. Remainder on division by 7: 2
[itex]2^8= 256[/itex]. Remainder on division by 7: 4
Get the idea?