To find the units digit of the expression $211^{870} + 146^{127} \times 3^{424}$, we need to find the units digit of each term separately and then combine them.

Units Digit of $211^{870}$

The units digit of $211^{870}$ depends only on the units digit of the base, which is 1. Any positive integer power of a number ending in 1 always results in a number ending in 1.

• Units digit of $211^{870}$ is 1.

Units Digit of $146^{127}$

The units digit of $146^{127}$ depends only on the units digit of the base, which is 6. Any positive integer power of a number ending in 6 always results in a number ending in 6.

• Units digit of $146^{127}$ is 6.

Units Digit of $3^{424}$

The units digit of $3^{424}$ depends on the pattern of the units digits of powers of 3:

• $3^1 = 3$
• $3^2 = 9$
• $3^3 = 27 \implies 7$
• $3^4 = 81 \implies 1$
• $3^5 = 243 \implies 3$

The pattern of the units digits of powers of 3 is (3, 9, 7, 1), which repeats every 4 powers.

To find the units digit of $3^{424}$, we examine the exponent 424 modulo 4:

$ 424 \pmod{4} = 0 $

Since the remainder is 0, the units digit corresponds to the last digit in the cycle (which is the 4th digit), which is 1.

• Units digit of $3^{424}$ is 1.

Units Digit of Product Term ($146^{127} \times 3^{424}$)

To find the units digit of the product, we multiply the units digits of $146^{127}$ and $3^{424}$:

Units digit of ($146^{127} \times 3^{424}$) = Units digit of (Units digit of $146^{127}$ $\times$ Units digit of $3^{424}$)

Units digit of ($146^{127} \times 3^{424}$) = Units digit of ($6 \times 1$)

Units digit of ($146^{127} \times 3^{424}$) = 6

Final Units Digit Calculation

Now, we find the units digit of the sum by adding the units digits of the two main parts:

Units digit of ($211^{870} + 146^{127} \times 3^{424}$) = Units digit of (Units digit of $211^{870}$ + Units digit of $146^{127} \times 3^{424}$)

Units digit of ($211^{870} + 146^{127} \times 3^{424}$) = Units digit of ($1 + 6$)

Units digit of ($211^{870} + 146^{127} \times 3^{424}$) = 7