# CMM Level E Assignments

### Charlotte Math Meetup Assignments (Level E)

These problems can be solved by elementary school math. They are designed from CMM Level E students, though students in Level M may find them interesting too.

Note: The answer to each problem should be a number. Please also report how many minutes it takes you to finish each problem. Do not rush.

**7th Assignment (Due: 8:00pm ET, May 3, 2020)**

1. N is a natural number. There are 12 common factors between 1,000,000,000,000 and N. What's the smallest possible value for N?

2. Leo and his little sister Angela are working on a science project together. They use pipettes to drop liquid into a 0.7 liter container. It takes Angela 24 seconds to fill in a 10 cm^{3} tube. Leo can do this 3 times as fast as Angela. The how many minutes does it take them to fill in the container?

3. Leo has a double dutch rope that is 16 feet long. He is using it to wrap rectangles with each side as a whole number. Each time he may or may not use the entire rope. How many rectangles with differet shapes can he create using this long rope?

4. Kyle is playing Pokémon Go while walking from home to library with his dad. To maximize the distance Pokémon Go records, Kyle decides to run at 4m/s towards the library the moment they leave home; once he arrives at the library, he turns around and runs at the same speed towards his dad; once he meets his dad, he turns around and runs at the same speed towards the library again; until both Kyle and his dad arrive at the library. If the distance between his home and library is 500m, and Kyle’s dad is walking at 2m/s, how many meters does Kyle run from the moment they leave home until they both arrive at the library?

**6th Assignment (Due: 8:00pm ET, April 26, 2020)**

1. If we define A#B = AB+A+B, what is 1#9#9#9#9#1?

2. The perfect squares are the squares of the whole numbers. How many perfect squares are there between 20 and 2020?

3. Find the 2020^{th} digit after decimal point of 2020/7.

4. A, B, C, D, E, F, and G are consecutive prime numbers arranged in the increasing order. If the sum of them has a prime factor 2, what is the product of them?

**5th Assignment (Due: 8:00pm ET, April 19, 2020)**

1. Every week, Leo works on the 7 problems from CMM assignments. During the 3rd week, he recorded the time for each problem, and found the following interesting facts: problems 1, 2, 3 together took him 6 minutes; problems 2, 3, 4 together took him 7 minutes; problems 3, 4, 5 together took him 8 minutes; problems 4, 5, 6 together took him 9 minutes; problems 5, 6, 7 together took him 10 minutes; problems 1, 6, 7 together took him 11 minutes; problems 1, 2, 7 together took him 12 minutes. How many minutes did Leo spend on problem #5?

2. 48 CMM students have a meetup together. Each holds a square tile. They lay the tiles down to form a big 7 by 7 grid, with the center position empty. How many squares are there that are fully covered by tiles.

3. After the first year of meetups, Dr. Hong identifies 10 students who have the best record on their assignments. He decides to run a competition. He randomly divides the students into 5 teams, with each team having at least 1 student. Student(s) of the winning team will be recognized as the spotlight student(s) of the year. How many different ways can he form the 5 teams? (This problem is beyond Level E, so it is not counted against spotlight student eligibility.)

4. Dr. Hong has a magic number in his mind. He asks the CMM students to guess the number with the following clues:

1) The number is a 6-digit number ABACDD;

2) It has three and only three two-digit prime factors EF, BE, and GD;

3) The product of BE and GD is a four-digit number EGHF.

4) A, B, C, D, E, F, G, H represent 8 different single digit numbers.

what is Dr. Hong's magic number ABACDD?

**4th Assignment (Due: 8:00pm ET, April 12, 2020)**

1. When two people shake hands with each another, that counts as one handshake. After the COVID-19 quarantine, all students from Dr. Hong's Charlotte Math Meetup have a big party in person. Everyone shakes hands with each other exactly once. There are 1225 handshakes! How many students are in the party?

2. Terminal zeros of a number are the zeros to the right of the last nonzero digit. For instance, 506,400,000 has 5 terminal zeros. During a math meetup, students play a game. They form a queue. The first student yells 1, the second student yells 2, the third student yells 6, the fourth student yells 24, the nth student yells the result of 1 x 2 x 3 x 4 x ... x n. Leo is the last student to yell his number. He forgets to bring the calculator, but he says there are 7 terminal zeros in his number. If Leo's statement is true, what's the largest possible number of students in the queue?

3. Leo, Kyle, Issaic, and several other Charlotte Math Meetup students form a circle to play a game of counting numbers clockwise. Each round starts from Leo, with each student on the circle adds the same prime number continuously. In Round 1, Leo counts 2. When it's Leo's turn to begin Round 6, he counts 107. Then three more students join the circle to continue the same game. What is the number Leo counts in Round 10?

4. A, B, C, D represent four different non-zero digits. AB x CD = DDD. What's the largest possible value of D?

**3rd Assignment (Due: 8:00pm ET, April 5, 2020)**

1. The fraction 360/N in the simplest form is a whole number. N is a positive integer. What is the total number of different values that N can be?

2. Leo plans to publish his diary. The pages of his diary book are numbered consecutively starting from 1. The digit 4 is printed 44 times on page numbers. What's the minimum number of pages his diary book can have?

3. Dr. Hong is trying to group students in his meetup. If he forms groups of 2, one student is left out. If he forms groups of 3, one student is left out. If he forms groups of 4, one student is left out. If he forms groups of 5, one student is left out. If he forms groups of 6, how many students would be left out?

4. If we rearrange all digits of a three digit number from largest to smallest, we get ABC. Moreover, the original three digit number equals ABC - CBA. What is the original three digit number?