# Target Test Prep Sample Units Digit Pattern Questions

### Example 1

Which of the following is the units digit of 7^{49}?

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 7^{48} has a units digit of 1. Then, 7^{49} has a units digit of 7.

### Example 2

If x is a positive integer, what is the units digit of 3^{16x+18}?

We can simplify this fraction to 3^{16x+18} = 3^{16x} × 3^{18}. We know that the units digits of powers of 3 exhibit the four-number sequence 3–9–7–1. Let’s evaluate 3^{16x} and 3^{18} separately:

^{16x}: 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, 3^{16x} must have a 1 as the units digit (because 16x is a multiple of 4).

^{18}: The closest multiple of 4 before 3^{18} is 3^{16}. We see that 3^{16} has a units digit of 1. Thus, 3^{18} 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, 3^{16x }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.

### Example 3

Variables x and y are positive integers, and

#### Quantity A

3

#### Quantity B

The units digit of 7^{xy+y}

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

7^{1 }= 7, 7^{2 }= 49, 7^{3 }= 343,7^{4 }= 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 7^{xy+y}. It may be helpful to express 7^{xy+y} as (7^{xy}) × 7^{y}.

We know that ^{multiple of 8}) × (7^{multiple of 8}), and because 8 is a multiple of 4, we can further simplify our thinking to (7^{multiple of 4}) × (7^{multiple 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 7^{xy+y }is 1 × 1 = 1.

Thus, Quantity A is greater than Quantity B.