Unit Digit Calculation for $\$(623)^{36}\$

To find the unit digit (also known as the unit place digit or the last digit) of a number raised to a power, we only need to focus on the unit digit of the base number. The other digits in the base do not affect the unit digit of the final result.

Unit Digit Concept Explained

For the expression $\$(623)^{36}\$$, the base is 623. The unit digit of the base is 3. Therefore, to find the unit digit of $\$(623)^{36}\$$, we simply need to find the unit digit of $\$(3)^{36}\$.

Cyclicity of Unit Digits for Base 3

The unit digits of powers of any single digit follow a repeating pattern, which is called cyclicity. Let's observe the pattern of the unit digits for powers of 3:

• $\$(3)^1 = 3\$ \quad \text{(Unit digit is 3)}$
• $\$(3)^2 = 9\$ \quad \text{(Unit digit is 9)}$
• $\$(3)^3 = 27\$ \quad \text{(Unit digit is 7)}$
• $\$(3)^4 = 81\$ \quad \text{(Unit digit is 1)}$
• $\$(3)^5 = 243\$ \quad \text{(Unit digit is 3)}$

As you can see, the unit digits repeat in a cycle of 3, 9, 7, 1. The length of this cycle is 4. This means that every 4th power brings the unit digit back to 1, and the pattern restarts.

Unit Digit Application for $\$(623)^{36}\$

To find the unit digit of $\$(3)^{36}\$$, we divide the exponent, which is 36, by the length of the cycle, which is 4. The remainder of this division will tell us which position in the cycle the unit digit falls on.

Calculation:

\$36 \div 4 = 9\$ with a remainder of \$0\$.

When the remainder is 0, it means the unit digit is the last digit in the cycle. In this case, the 4th digit in the cycle of 3 (which is 3, 9, 7, 1) is 1. If the remainder were 1, the unit digit would be the 1st in the cycle (3); if 2, the 2nd (9); if 3, the 3rd (7).

Unit Digit Conclusion for the Expression

Since the remainder of 36 divided by 4 is 0, the unit digit of $\$(3)^{36}\$$ is the same as the unit digit of $\$(3)^4\$$, which is 1.

Therefore, the unit digit in $\$(623)^{36}\$$ is 1.