The last (ones) digit of 3^{2} is 9 since 3 x 3 = **9**

The last (ones) digit of 3^{3} is 7 since 3 x 3 x 3 = **27**

The last (ones) digit of 3^{4} is 1 since 3 x 3 x 3 x 3 = **81.**

What is the ones digit of 3^{2019}?

Do similar patterns exist in the powers of other whole numbers? For example, what is the ones digit of each of the following?

- 5
^{2019} - 2
^{2019} - 4
^{2019} - 8
^{2019} - 7
^{2019}

Digital technology allows for powers to be calculated easily. Since the purpose of the task is for students to identify and apply sequential patterns, rather than perform mental or written calculations, it is appropriate that digital technology is freely available. For example, students can be asked to program a spreadsheet to calculate the powers of three. They might begin with a layout like this. Note that a recursive formula as shown will take the cell content above and multiply that number by three.

A limitation is that most spreadsheets limit the number of digits available in a cell to 15 or 16. That will mean that the display below occurs:

The limitation is good in that students will need to predict the pattern. Students should identify that the ones digits follow a repeating sequence 1, 3, 9, 7, 1, 3, 9, 7, …

Students can use their scientific calculator to though the same restrictions apply to the length of the number displayed. For example, 3^{20} will display as 3486784401. However, 321 will not display correctly as it requires 11 digits. So even with support of digital technology students need to apply a conceptual approach to find the ones digit of 3^{2019}.

Reasoning conceptually about a repeating pattern involves identifying the element of repeat and using that element to predict further members in the pattern. Consider the powers of three:

1, 3, 9, 27, 81, 243, 729, 2187, …

The pattern of repeat in the ones digit is 1, 3, 9, 7 so every fourth power of two the pattern recurs. Using divisibility of the exponent by four is the most efficient way to predict the ones digit in powers of three. If the exponent is a multiple of four then the ones digit of the power is 1.

Expect students to reason with divisibility to establish a way to find the ones digit of 32019.

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Therefore, 3^{2019} will have 7 as the ones digit because 3^{2020} will have 1 as the ones digit, since 2020 is divisible by four (remainder zero).