Target Test Prep Sample Units Digit Pattern Questions

Example 1

medium

Which of the following is the units digit of 3247?

1

3

5

7

9

Confirm your answer

Solution:

The pattern for the units digits of powers of 3 is 3–9–7–1. This means that all powers of 3 with a multiple of 4 in the exponent will have a 1 as the units digit. A multiple of 4 that is close to 247 is 244. This means that the units digit of 3244 is a 1, that of 3245 is a 3, that of 3246 is a 9, and that of 3247 is a 7.

Correct answer:D

Example 2

hard

If x and y are positive integers, what is the units digit of 7xy+y?

1) x16=n, where n is a positive integer.

2) y8=m, where m is a positive integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
Confirm your answer

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 7xy+y. It may be helpful to simplify 7xy+y to (7xy)×(7y).

Statement One Alone:

x16=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:

y8=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 (7multiple of 8)×(7multiple of 8), and because 8 is a multiple of 4, we can further simplify our thinking to (7multiple of 4)×(7multiple of 4). We know that all powers of 7 that are multiples of 4 have a units digit of 1. Thus, the units digit of x=t2 is 1 × 1 = 1. Statement two alone is sufficient.

Correct answer:B

Example 3

medium

202 + 212 + 222 + 232 + 242 + 252 =

3,055

3,060

3,066

3,704

3,077

Confirm your answer

Solution:

Since each answer choice has a different units digit, instead of finding the actual sum we can just find the units digit of the sum. Let's use the units digits of each square to determine the units digit of the sum.

⇒ The units digit of 202 must be 0, since 02 = 0.

⇒ The units digit of 212 must be 1, since 12 = 1.

⇒ The units digit of 222 must be 4, since 22 = 4.

⇒ The units digit of 232 must be 9, since 32 = 9.

⇒ The units digit of 242 must be 6, since 42 = 16.

⇒ The units digit of 252 must be 5, since 52 =25.

With this, we can sum the units digits: 0 + 1 + 4 + 9 + 6 + 5 = 25. Thus, the units digit is 5.

Correct answer:A