### Theory:

We have learnt how to find the unit digits of base ending with the numbers $$0$$, $$1$$, $$4$$, $$5$$, $$6$$ and $$9$$.

We shall learn how to find the unit digits of the base ending with the numbers $$2$$, $$3$$, $$7$$ and $$8$$.

Let us observe the pattern for the base ending with the number $$2$$.
Pattern for the number $$2$$:

$$2^1 = 2$$

$$2^2 = 2 \times 2 = 4$$

$$2^3 = 2 \times 2 \times 2 = 8$$

$$2^4 = 2 \times 2 \times 2 \times 2 = 16$$

$$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$$

Here, in the above pattern, we can see that $$2^5$$ and $$2^1$$ has the same unit digit $$2$$. Therefore, after every $$4^{th}$$ multiplication, the unit digit will be $$2$$.

Pattern for the number $$3$$:

$$3^1 = 3$$

$$3^2 = 3 \times 3 = 9$$

$$3^3 = 3 \times 3 \times 3 = 27$$

$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

$$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$$

We can see that $$3^5$$ and $$3^1$$ has the same unit digit $$3$$. Therefore, after every $$4^{th}$$ multiplication, the unit digit will be $$3$$.

Pattern for the number $$7$$:

$$7^1 = 7$$

$$7^2 = 7 \times 7 = 49$$

$$7^3 = 7 \times 7 \times 7 = 343$$

$$7^4 = 7 \times 7 \times 7 \times 7 = 2401$$

$$7^5 = 7 \times 7 \times 7 \times 7 \times 7 = 16807$$

Thus, in $$7^5$$ and $$7^1$$ has the same unit digit $$7$$. Therefore, after every $$4^{th}$$ multiplication, the unit digit will be $$7$$.

Pattern for the number $$8$$:

$$8^1 = 8$$

$$8^2 = 8 \times 8 = 64$$

$$8^3 = 8 \times 8 \times 8 = 512$$

$$8^4 = 8 \times 8 \times 8 \times 8 = 4096$$

$$8^5 = 8 \times 8 \times 8 \times 8 \times 8 = 32768$$

Thus, in $$8^5$$ and $$8^1$$ has the same unit digit $$8$$. Therefore, after every $$4^{th}$$ multiplication, the unit digit will be $$8$$.
Example:
1. Find the unit digit of the number $$2^{294}$$.

Solution:

Let us divide the power $$294$$ by $$4$$. We have the remainder as $$2$$.

That is, in the pattern, $$2^2 = 2 \times 2 = 4$$.

Hence, the unit digit is $$4$$.

2. Find the unit digit of the number $$3^{596}$$.

Solution:

Let us divide the power $$596$$ by $$4$$. We have the remainder as $$0$$.

Since, after every $$4^{th}$$ multiplication, the unit digit will be $$3$$. Thus, when the remainder is $$0$$, our answer should be $$4^{th}$$ power.

That is, $$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

Hence, the unit digit is $$1$$.