**Question 8:**If 100 positive integers are randomly selected, at least how many of them are guaranteed to have the same remainder when divided by 7?**Question 9:**At least how many positive integers must be selected in order to guarantee:(a) At least two of them have the same remainder when divided by 7?(b) At least five of them have the same remainder when divided by 7?

8:The problem is related to the Pigeonhole Principle. The Pigeonhole Principle states that if we distribute n items into m containers and,n>m⋅k then at least one container must contain more than k items.In this case, we have 100 positive integers and 7 possible remainders when divided by 7 (0, 1, 2, 3, 4, 5, 6). Each remainder corresponds to a "container."To guarantee that at least two of the integers have the same remainder when divided by 7, we can use the Pigeonhole Principle. Since we have 100 integers and 7 possible remainders, we can distribute the integers into the remainders:n>m⋅k100>7⋅kk<1007≈14.29So, we need at least 15 positive integers to guarantee that at least two of them have the same remainder when divided by 7.9 a:This problem involves the Pigeonhole Principle. The Pigeonhole Principle states that if we distribute n items into m containers and n>m⋅k, then at least one container must contain more than k items.In this case, the "pigeons" are the positive integers, the "pigeonholes" are the possible remainders when divided by 7 (0, 1, 2, 3, 4, 5, 6), and the "containers" are these remainders.a. At least two of them have the same remainder when divided by 7:We need to find the minimum number of positive integers (n) to guarantee that at least two of them have the same remainder when divided by 7. This corresponds to the minimum number of pigeons needed to guarantee that at least one pigeonhole contains more than one pigeon.Using the Pigeonhole Principle:n>7So, at least 8 positive integers must be selected to guarantee that at least two of them have the same remainder when divided by 7.b. At least five of them have the same remainder when divided by 7:Now, we need to find the minimum number of positive integers (n) to guarantee that at least five of them have the same remainder when divided by 7.n>7⋅4n>28So, at least 29 positive integers must be selected to guarantee that at least five of them have the same remainder when divided by 7.