# Number Theory

## Odds and Evens

The sum of any four consecutive odd integers is a multiple of eight. For example, 1 + 3 + 5 + 7 = 16 and 5 + 7 + 9 + 11 = 32. What is the product of the greatest and least of four consecutive odd integers whose sum is 64?

## Happy New Year!

Happy New Year! Now that 2019 is here, let’s have some fun solving this collection of problems that all involve the number 2019.

What is the greatest prime factor of 2019?

*The factors of 2019 are 1, 3, 673, 2019. The prime factors of 2019 are 3 and 673, the greatest being ***673***.*

## Best of 2018

As 2018 comes to an end, let’s go back and solve some of our favorite problems of year.

__School Handbook__

**Problem 237: **Kendra starts at a positive integer *k *and counts up by 4s until she hits exactly 200. Mason starts at a positive integer *m *and counts up by 6s until he hits exactly 200. If it takes Kendra half as many steps to reach 200 as it takes Mason, what is the greatest possible value of *k *− *m*?

## Counting Cookies

Santiago baked gingerbread cookies in the shapes of children. He gave each gingerbread boy and girl eyes made from chocolate covered candies in one of four different colors, a nose made from a butterscotch chip, toffee chip or peanut butter chip, and a mouth made from a piece of red licorice. Santiago then gave each gingerbread boy and girl buttons made from gum drops in one of four different flavors. For the final detail, he gave each gingerbread boy a blue bow tie, and each gingerbread girl a red hair bow.

## Prime Number Sense

If *n* is the sum of three consecutive primes and is also the product of two 2-digit primes, what is the least possible value of *n*?

*Let’s try the two 2-digit primes of least value, 11 and 13. These primes have a product of 143, and 143/3 is around 47. Two primes close in value to 47 are 43 and 53. So we try the sum 43 + 47 + 53 and see that it does, in fact, equal 143.*

## Making Sense of Numbers

Here are a few problems to keep your number sense sharp during the summer months.

What is the greatest positive three-digit integer that is divisible by 5, 7 and 9?

*We know that any three-digit number that is divisible by 5, 7 and 9, is divisible by 5 × 7 × 9 = 315. The largest three-digit multiple of 315 is 315 × 3 = 945.*

## We're #ALLCAPS

Congratulations to the Washington Capitals hockey team on winning the Stanley Cup! To celebrate the NHL champions, we have a few hockey themed problems for you to solve.

In the quest to win the Stanley Cup, the Washington Capitals played in four playoff rounds. During each round, they faced a different opponent in a best-of-seven series, in which the winner was the first team to win four games. What is the absolute difference between the greatest and least number of games that a team could play in four such rounds to become the Stanley Cup Champions?

## Graduation Gift Cards

Derrick received a total of 21 gift cards for graduation. He received gift cards in each of four denominations, $25, $50, $75 and $100. The number of $100 gift cards Derrick got was twice the number of $75 gift cards he received. The total value of the $50 gift cards he got exceeded the total combined value of the $25, $75 and $100 gift cards by $100. How many $25 gift cards did Derrick get for graduation?

## 2018 National Champion

Last week the National Competition concluded, and Luke Robitaille from Texas became the first ever two-time MATHCOUNTS National Champion. Let’s look at some of the problems he had to solve on the way to the top!

## National Competition

On Sunday, May 13th, 224 of the nation’s smartest middle-school math minds will be in Washington, DC for the 2018 Raytheon MATHCOUNTS National Competition. The stakes are high, and the problems will be tough. Here are a few Sprint Round problems national competitors solved in 2017.

What is the maximum possible absolute difference between a two-digit integer and the two-digit integer resulting when the digits are reversed?

*[Sprint #2] *