Target Test Prep Sample Units Digit Pattern Questions

Solution:

Question Stem Analysis:

The units digits of powers of 7 follow the four-number pattern 7–9–3–1. We need to determine the units digit of $7^{\mathrm{x}\mathrm{y}+\mathrm{y}}$. It may be helpful to simplify $7^{\mathrm{x}\mathrm{y}+\mathrm{y}}$ to $\left(7^{\mathrm{x}\mathrm{y}}\right)\times \left(7^{\mathrm{y}}\right)$.

Statement One Alone:

⇒ $\frac{\mathrm{x}}{16}=\mathrm{n}$, where n is a positive integer.

We can restate this as x = 16n. Because n is an integer, it must be true that x is a multiple of 16. This means that xy is a multiple of 16 because the product of 16 and any positive integer is a multiple of 16. Because 16 is a multiple of 4, we also know that xy is a multiple of 4. However, because we do not know anything about y, statement one alone is not sufficient.

Eliminate answer choices A and D.

Statement Two Alone:

⇒ $\frac{\mathrm{y}}{8}=\mathrm{m}$, where m is a positive integer.

We can restate this as y = 8m. Because m is an integer, it must be true that y is a multiple of 8. This also means that xy is a multiple of 8. We can now think about this as $\left(7^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\text{ of 8}}\right)\times \left(7^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\text{ of 8}}\right)$, and because 8 is a multiple of 4, we can further simplify our thinking to $\left(7^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\text{ of 4}}\right)\times \left(7^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\text{ of 4}}\right)$. We know that all powers of 7 that are multiples of 4 have a units digit of 1. Thus, the units digit of $\mathrm{x}={\mathrm{t}}^2$ is 1 × 1 = 1. Statement two alone is sufficient.