Problem of the Week Archive
Happy New Year! – January 4, 2016
Problems & Solutions
Happy New Year! Now that 2016 is here, let’s have some fun with this number!
What is the last digit of 20162016?
The units digit of 20162016 will only depend on the units digit of the previous power. For example, the units digit of 20162 will
be the same as the units digit of 62, and the units digit of 20163 will be the same as the units digit of 63, and so forth. So we
are looking for a pattern when computing powers of 6. The value of 62 = 36, the value of 63 = 216, since the last digit is a
6 multiplied by a 6, no matter what power 2016 is raised to, the last digit will remain 6.
What is the sum of the first 2016 positive integers?
Certainly we don’t have time to add all of these numbers to determine the sum. If we were to list all 2016 integers, it would look
like this: 1, 2, 3, 4, …, 2013, 2014, 2015, 2016. Notice that the 1st and 2016th terms have a sum of 2017, as do the 2nd
and 2015th, and so on. That means there are 2016/2 = 1008 sums of 2017. Therefore, the sum of the first 2016 positive
integers is 1008 x 2017 = 2,033,136.
What is the area of an equilateral triangle with perimeter 2016 units? Express your answer in
simplest radical form.
Because the triangle is equilateral and the perimeter is 2016 units, the length of each side is 2016 ÷ 3 = 672 units. The
altitude of the triangle is 672 ÷ 2 × √3 = 336 √3 units. The area of the triangle is 1/2 × 672 × 336 √3 = 112,896 √ 3 units2.
Problem of the Week Archive
Happy New Year! – January 4, 2016
Problems
Happy New Year! Now that 2016 is here, let’s have some fun with this number!
What is the units digit of 20162016?
What is the sum of the first 2016 positive integers?
What is the area of an equilateral triangle with perimeter 2016 units? Express your answer in
simplest radical form.