 Number Theory

# GCD / LCM Warmup If one person claps every other beat and another person claps every 3rd beat, what is the first beat they will both clap? Hot dogs are typically sold in packages of 10, while hot dog buns are typically sold in packages of 8. Suppose you want to buy exactly the same number of hot dogs as hot dog buns. What is the minimium number of hot dogs you will have to buy?

Suppose you want to tile a rectangular floor that measures 132 ft. by 150 ft. using square tiles that are all the same size, with no overlap and no squares hanging over the edge. What are the largest square tiles you can use? If you place $$27 \times 27$$ square tiles alongside $$12 \times 12$$ square tiles as shown in the figure above until the first time the edge of a $$27 \times 27$$ tile lines up with the edge of a $$12 \times 12$$ tile, how many $$12 \times 12$$ tiles will you end up using? If one person claps every 3rd beat and another person claps every 4th beat, will they ever clap at the same time?

×