What is the thousands digit of 9^1000?
To determine the thousands digit of 9^1000, we need to evaluate the value of 9^1000 and then extract the thousands digit. However, calculating such a large number manually would be very time-consuming. Instead, we can use a shortcut method.
First, let's observe the pattern of the thousands digit as we increase the powers of 9:
9^1 = 9 (thousands digit = 0)
9^2 = 81 (thousands digit = 0)
9^3 = 729 (thousands digit = 0)
9^4 = 6561 (thousands digit = 6)
9^5 = 59049 (thousands digit = 5)
9^6 = 531441 (thousands digit = 5)
9^7 = 4782969 (thousands digit = 4)
From this pattern, we can see that the thousands digit repeats every 6 powers. So, to find the thousands digit of 9^1000, we need to find the remainder when 1000 is divided by 6.
Dividing 1000 by 6 gives a quotient of 166 with a remainder of 4. Therefore, the thousands digit of 9^1000 will be the same as the thousands digit of 9^4, which is 6.
Hence, the thousands digit of 9^1000 is 6.