Target Test Prep Sample Units Digit Pattern Questions

Example 1

medium

Which of the following is the units digit of 749?                

1

3

5

7

9

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Solution:

Turning back to the units-digit matrix, we find that the units-digit pattern for powers of 7 is 7–9–3–1. Thus, all powers of 7 that are multiples of 4 have a units digit of 1. The closest multiple of 4 to 49 is 48. This means that 748 has a units digit of 1. Then, 749 has a units digit of 7.

Correct answer:D

Example 2

hard

If x is a positive integer, what is the units digit of 316x+18?

9

7

3

2

1

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Solution:

We can simplify this fraction to 316x+18 = 316x × 318. We know that the units digits of powers of 3 exhibit the four-number sequence 3–9–7–1. Let's evaluate 316x and 318 separately:

⇒ 316x: Every power of 3 that is a multiple of 4 has a units digit of 1. Because the product of 16 and any positive integer is a multiple of 16, it is also a multiple of 4. Thus, 316x must have a 1 as the units digit (because 16x is a multiple of 4).

⇒ 318: The closest multiple of 4 before 318 is 316. We see that 316 has a units digit of 1. Thus, 318 has a units digit of 9 (in the 3, 9, 7, 1 pattern, move two places from 1). This means that the product of this multiplication must have a units digit of 1 × 9 = 9.

A second way to solve this problem is to realize that because the product of 16 and any positive integer is a multiple of 16, it is also a multiple of 4. Thus, 316x must have a 1 as the units digit. Then we move forward 18 times in the pattern of 3, 9, 7, 1. Four moves in the pattern get us to 1, 8 moves get us to 1, 12 moves get us to 1, 16 moves get us to 1, and then 2 more moves bring us to the units digit 9.

Correct answer:A

Example 3

hard

Variables x and y are positive integers, and  y8=m, where m is a positive integer.

Quantity A

3

Quantity B

The units digit of 7xy+y

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
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Solution:

We first need to determine the units pattern of the powers of 7:

71 = 7, 72 = 49, 73 = 343,74 = 2401. From here we see 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 express 7xy+y  as (7xy) × 7y.

We know that y8=m , and 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 7xy+y is 1 × 1 = 1.

Thus, Quantity A is greater than Quantity B.

Correct answer:A