Question

Rick was counting his collection of bottle caps. He started organizing them in piles. He put all of the bottle caps in piles of four, but found that there was one left over. Then he put them in piles of five and found that there were two left over. Then he put them in piles of seven, and found that there was one left over. Finally he put them in piles of three, and there weren't any left over. He has less than 100 bottle caps. How many bottle caps does Rick have?

Answer 1

Rick has 58 bottle caps, which is the number that satisfies all the conditions given in the modular arithmetic problem.

Step-by-step explanation:

The student's question involves solving a classic type of problem in mathematics known as a 'Chinese Remainder Theorem' problem, which falls under the category of number theory. The problem has a series of modular arithmetic conditions.

Rick has a certain number of bottle caps, which when divided by 4 leaves a remainder of 1, when divided by 5 leaves a remainder of 2, when divided by 7 leaves a remainder of 1, and when divided by 3 leaves no remainder. To find the solution (the total number of bottle caps), we need to find a number less than 100 that satisfies all these conditions simultaneously.

By trial and error or systematic calculation, we find that the number 58 satisfies all the given conditions. It gives a remainder of 1 when divided by 4 (58%4 = 1), a remainder of 2 when divided by 5 (58%5 = 2), a remainder of 1 when divided by 7 (58%7 = 1), and no remainder when divided by 3 (58%3 = 0).