Problem of the Week Archive
Fun with 2017 – January 9, 2017
Problems & Solutions
The sum of the digits of 2017 is 10. What is the next
year in which this will occur?
The sum of the digits of 2017 is 10. Each consecutive year the
sum will increase by 1 until we reach 2020. The sum of the
digits in 2020 is 2 + 2 = 4. We need to add 6 to this to get to
a sum of 10, so the next year the sum of the digits will be 10 is
2020 + 6 = 2026.
The number 2017 is prime. What was the most recent
year before 2017 that was also a prime number?
Working backwards from 2017, we can skip all even numbered years since they will all be divisible by 2. So we
just need to look at odd number years. The first odd year before 2017 was 2015, which we know is divisible by 5
because it ends in the digit 5. The next odd year to check is 2013, which we know is divisible by 3 because its
digits sum to a multiple of 3. The next odd year is 2011, which is, in fact, prime. So the most recent prime year
was 2011.
The number 2017 can be expressed as the difference of the squares of two consecutive whole
numbers. What is the sum of these two numbers?
Let’s call the two numbers a and b. We can write a2 – b2 = 2017. The difference of squares can be factored to be
(a – b)(a + b) = 2017. Since we know that a and b are consecutive numbers, that means that a – b = 1.
Therefore, a + b = 2017.
Problem of the Week Archive
Fun with 2017 – January 9, 2017
Problems
The sum of the digits of 2017 is 10. What is the next year in
which this will occur?
The number 2017 is prime. What was the most recent year
before 2017 that was also a prime number?
The number 2017 can be expressed as the difference of the
squares of two consecutive whole numbers. What is the
sum of these two numbers?