3.1.5 Finding Units Digit of $x^n$

This is a common problem on the number sense test which seems considerably difficult, however there is a shortcut method. Without delving too much into the modular arithmetic required, you can think of this problem as exploiting patterns. For example, let’s find the units digit of $3^{47}$, knowing:

• $3^1 = 3$ (Units Digit: 3)
• $3^2 = 9$ (Units Digit: 9)
• $3^3 = 27$ (Units Digit: 7)
• $3^4 = 81$ (Units Digit: 1)
• $3^5 = 243$ (Units Digit: 3)
• $3^6 = 729$ (Units Digit: 9)
• $3^7 = 2187$ (Units Digit: 7)
• $3^8 = 6561$ (Units Digit: 1)

From this you can see the units digit repeats every 4th power.

So in order to see what is the units digit you can divide the power in question by 4 then see what the remainder $r$ is. And in order to find the appropriate units digit, you’d then look at the units digit of $3^r$. For example, the units digit for $3^5$ could be found by saying $5 \div 4$ has a remainder of 1 so, the units digit of $3^5$ corresponds to that of $3^1$ which is 3. So to reiterate, the procedure is:

1. For low values of $n$, compute what the units digit of $x^n$ is.
2. Find out how many unique integers there are before repetition (call it $m$).
3. Find the remainder when dividing the large $n$ value of interest by $m$ (call it $r$).
4. Find the units digit of $x^r$, and that’s your answer.

So for our example of $3^{47}$: $47 \div 4$ has a remainder of 3 $3^3$ has the units digit of 7

Other popular numbers of interest are:

Numbers Ending in	Repeating Units Digits	Number of Unique Digits
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
4	4, 6	2
5	5	1
6	6	1
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4
9	9, 1	2

Using the above table, we can calculate the units digit of any number raised to any power relatively simple. To show this, find the units digit of $27^{63}$:

From the table, we know it repeats every 4th power, so: $63 \div 4 \Rightarrow r = 3$ $r = 3$ corresponds to $7^3$ which ends in a 3.

This procedure is also helpful with raising the imaginary number $i$ to any power. Remember from Algebra:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$
• $i^5 = i$
• $i^6 = -1$
• $i^7 = -i$
• $i^8 = 1$

So, after noticing that it repeats after every 4th power, we can compute for example $i^{114}$. $114 \div 4$ has a remainder of $2 \Rightarrow i^2 = -1$

The following are examples of these types of problems:

Problem Set 3.1.5