find the 10 digits of 3^2007.
how many 4-digit numbers whose digits are all odd are multiples of 11?
I will use Congruences and Euler's Generalization of Fermat's Little Theorem..
We note that,
And,
Thus,
Bring both sides to the power of 50 thus,
Note that,
Thus, their multiplication yields,
Since it is congruent to 100 it shows the last two digits. The second to last digits it 8.
The smallest 4 digit number divisible by 11 is 91*11, and the largest
4 digit number divisible by 11 is 909*11. Therefore there are 819 such
numbers divisible by 11. Of these (819-1)/2 are even and the rest are odd
so there are 410 odd numbers 4 digit divisible by 11.
RonL