The set A consists of nine positive integers, none of which has a prime divisor larger than six. Prove that A is guaranteed to contain two distinct elements whose product is the square of an integer. Hint: Any positive integer that does not have a prime divisor larger than 6 can be written as 2^x1 3^x2 5^x3 for integers x1, x2, x3.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
The set A consists of nine positive integers, none of which has a prime divisor larger than six. Prove that A is guaranteed to contain two distinct elements whose product is the square of an integer.
Hint: Any positive integer that does not have a prime divisor larger than 6 can be written as 2^x1 3^x2 5^x3 for integers x1, x2, x3.
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