MarkFL said:I found this to be a very interesting problem...(Clapping)
MarkFL said:My solution:
I began by writing:
\(\displaystyle \frac{10^{20000}}{10^{100}+3}=10^{19900}-3\cdot\frac{10^{19900}}{10^{100}+3}\)
Continuing in this manner, we will find:
\(\displaystyle \frac{10^{20000}}{10^{100}+3}= \sum_{k=1}^{199}\left((-3)^{k-1}10^{100(200-k)} \right)-(3)^{199}+\frac{9^{100}}{10^{100}+3}\)
Hence:
\(\displaystyle \left\lfloor \frac{10^{20000}}{10^{100}+3} \right\rfloor=\sum_{k=1}^{199}\left((-3)^{k-1}10^{100(200-k)} \right)-(3)^{199}\)
\(\displaystyle \left\lfloor \frac{10^{20000}}{10^{100}+3} \right\rfloor=m\cdot10^{100}-(3)^{199}\) where \(\displaystyle m\in\mathbb{N}\)
Now, we need only find the units digit of \(\displaystyle 3^{199}\) and subtract it from 10. Borrowing from my solution to last week's High School POTW...
Observing that:
\(\displaystyle 3^{4(1)-1}=27\)
\(\displaystyle 3^{4(2)-1}=2187\)
We may choose to state the induction hypothesis $P_n$:
\(\displaystyle 3^{4n-1}=10k_n+7\)
As the induction step, we may add:
\(\displaystyle 3^{4(n+1)-1}-3^{4n-1}=80\cdot3^{4n-1}=80\left(10k_n+7 \right)\)
to get:
\(\displaystyle 3^{4(n+1)-1}=80\left(10k_n+7 \right)+10k_n+7=10\left(8\left(10k_n+7 \right)+k_n \right)+7\)
If we make the recursive definition:
\(\displaystyle k_{n+1}\equiv8\left(10k_n+7 \right)+k_n\) where \(\displaystyle k_1=2\)
we then have:
\(\displaystyle 3^{4(n+1)-1}=10k_{n+1}+7\)
We have derived $P_{n+1}$ from $P_n$ thereby completing the proof by induction.
Thus, we find the units digit of the original expression is:
\(\displaystyle 10-7=3\)
To find the units digit of an expression, you can follow these steps:1. Simplify the expression as much as possible.2. Look at the units digit of each number in the expression.3. Use basic arithmetic operations (addition, subtraction, multiplication, and division) to find the units digit of the simplified expression.4. The units digit of the simplified expression will be the units digit of the original expression.
No, the units digit of an expression can never be negative. The units digit is always a positive integer between 0 and 9.
Yes, there are some common patterns and rules that can help you find the units digit of an expression more easily. For example, the units digit of the sum of two numbers will be the same as the units digit of their sum. Also, the units digit of a number multiplied by itself multiple times will always be the same as the units digit of the original number. These patterns can vary depending on the specific expression, but they can be helpful to keep in mind.
Yes, you can use a calculator to find the units digit of an expression. However, it is important to make sure the calculator is set to display the correct number of digits. Some calculators may round the result, which can affect the units digit of the expression.
Yes, finding the units digit of an expression can be important in some cases. It can help you verify the accuracy of your calculations and can also be useful in solving more complex math problems. Additionally, in some fields such as computer science, knowing the units digit is necessary for certain calculations and algorithms.