# Number Theory

## 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] *

## Springtime Babies

A baby rabbit moves 5 inches forward with each hop and his mother moves 14 inches forward with each hop. After the mother and baby each hop 10 hops in the same direction, the mother stops and waits for her baby to catch up. How many hops does the baby have to take to catch up?

## Get Ready for State

This month the 2018 MATHCOUNTS State Competitions will take place, so let’s look back at some of the 2017 State Sprint round problems and solve them in preparation.

Andre can complete 5/6 of a job in 3/4 of the time that it takes Michael to do the whole job. What is the ratio of the rate at which Andre works to the rate at which Michael works? Express your answer as a common fraction.

## One for the Ages

Cara was born on January 1, 2010, and her mother, Sydney, was born on January 1, 1982. In what year will Sydney’s age be twice Cara’s age?

If we let Cara’s age be C, then Sydney’s age is S = C + 28, since Cara was born 28 years after Sydney. We are interested in determining when Sydney’s age is twice Cara’s age, in other words, when S = 2C. Substituting, we have C + 28 = 2C. Solving for C we get C = 28. Therefore, Sydney’s age will be twice Cara’s age in the year 2010 + 28 =.2038

## Happy New Year!

Happy New Year! Now that 2018 is here, let’s have some fun with this number!

What is the last digit of 2018^{2018}?

The units digits of the powers of 2018 are the same as the units digits of the powers of 8, and they form a pattern that repeats every four powers. The pattern of units digits is 8, 4, 2 and 6. Since 2018 is 2 more than a multiple of 4, it follows that the units digit of 2018^{2018}is the second digit in this pattern, which is.4