What is the units (ones) digit of the integer equal to $5^{2019}-3^{2019}$ ?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8
The units digit of $5^{2019}$ is 5 .

This is because the units digit of any power of 5 is 5 .

To see this, we note that the first few powers of 5 are

$$
5^{1}=5 \quad 5^{2}=25 \quad 5^{3}=125 \quad 5^{4}=625 \quad 5^{5}=3125 \quad 5^{6}=15625
$$

The units digit of a product of integers depends only on the units digits of the integers being multiplied. Since $5 \times 5=25$ which has a units digit of 5 , the units digits remains 5 for every power of 5 .

The units digit of $3^{2019}$ is 7 .

This is because the units digits of powers of 3 cycle $3,9,7,1,3,9,7,1, \ldots$

To see this, we note that the first few powers of 3 are

$$
3^{1}=3 \quad 3^{2}=9 \quad 3^{3}=27 \quad 3^{4}=81 \quad 3^{5}=243 \quad 3^{6}=729
$$

Since the units digit of a product of integers depends only on the units digits of the integers being multiplied and we multiply by 3 to get from one power to the next, then once a units digit recurs in the sequence of units digits, the following units digits will follow the same pattern. Since the units digits of powers of 3 cycle in groups of 4 and 2016 is a multiple of 4, then $3^{2016}$ has a units digit of 1 .

Moving three additional positions along the sequence, the units digit of $3^{2019}$ will be 7 .

Since the units digit of $5^{2019}$ is 5 and the units digit of $3^{2019}$ is 7 , then the units digit of the difference will be 8 . (This is the units digit of the difference whenever a smaller integer with units digit 7 is substracted from a larger integer with units digit 5.)

ANSWER: (E)

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