a) given any set of seven integers, must there be at least two that have the same remainder when divided by 6? to answer this question, let a be the set of 7 distinct integers and let b be the set of all possible remainders that can be obtained when an integer is divided by 6, which means that b has elements. hence, if a function is constructed from a to b that relates each of the integers in a to its remainder, then by the ---select--- principle, the function is ---select--- . therefore, for the set of integers in a, it is ---select--- for all the integers to have different remainders when divided by 6. so, the answer to the question is ---select--- . (b) given any set of seven integers, must there be at least two that have the same remainder when divided by 8? if the answer is yes, enter yes. if the answer is no, enter a set of seven integers, no two of which have the same remainder when divided by 8.

The remainder is the value left after the division. If a number is not completely divisible by another number then we are left with a value once the division is done. This value is called the Remainder.

Given any set of seven integers, there must be at least two numbers that have the same remainder when divided by 6.

So there can be six remainders when divided by 6 i.e. 0,1,2,3,4 and 5.

According to the Pigeonhole principle,

in any set of seven integers, two must have the same remainder when divided by seven.

Consider the set of integers 0,1,2,3,4,5 and 66. All of these have different remainders upon division by 8.

Hence there need not be two numbers such that they have the same remainders when divided by 6.

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