Unit Digit Pattern for Powers of 4

To find the unit digit of $(14)^{112} + (14)^{113}$, we first need to understand the pattern of the unit digits of powers of numbers ending in 4.

The unit digit of any power of a number depends only on the unit digit of the base number. In this case, the base number is 14, and its unit digit is 4.

Let's examine the unit digits of the first few powers of 4:

• $4^1 = 4$ (Unit digit is 4)
• $4^2 = 16$ (Unit digit is 6)
• $4^3 = 64$ (Unit digit is 4)
• $4^4 = 256$ (Unit digit is 6)

We can observe a repeating pattern for the unit digits of powers of 4: 4, 6, 4, 6, ...

This pattern follows a simple rule:

• If the exponent is an odd number, the unit digit is 4.
• If the exponent is an even number, the unit digit is 6.

Calculating Unit Digits for the Given Expression

Now, let's apply this rule to the terms in the given expression $(14)^{112} + (14)^{113}$.

• For the term $(14)^{112}$:
  • The exponent is 112, which is an even number.
  • Therefore, the unit digit of $(14)^{112}$ is 6.
• For the term $(14)^{113}$:
  • The exponent is 113, which is an odd number.
  • Therefore, the unit digit of $(14)^{113}$ is 4.

Finding the Unit Digit of the Sum

To find the unit digit of the sum $(14)^{112} + (14)^{113}$, we simply need to add the unit digits of the individual terms and find the unit digit of the result.

Unit digit of $(14)^{112}$ = 6

Unit digit of $(14)^{113}$ = 4

Sum of unit digits = $6 + 4 = 10$.

The unit digit of the sum (10) is 0.

Thus, the unit digit in $(14)^{112} + (14)^{113}$ is 0.