After you complete your survey of students reported heights at your school, you discover that your sample is biased. Exactly 10%
2 answers:
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Answer:
The family of all prime numbers such that is a perfect square is represented by the following solution:
is an arbitrary prime number. (1)
(2)
is another arbitrary prime number. (3)
Step-by-step explanation:
From Algebra we know that a second order polynomial is a perfect square if and only if . From statement, we must fulfill the following identity:
By Associative and Commutative properties, we can reorganize the expression as follows:
(1)
Then, we have the following system of equations:
(2)
(3)
(4)
By (2) and (4) in (3), we have the following expression:
From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, . If
,
and
are prime numbers, then
must be an even composite number, which means that
and
can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.
In addition, must be a natural number, which means that:
But the lowest possible product made by two prime numbers is . Hence,
.
The family of all prime numbers such that is a perfect square is represented by the following solution:
is an arbitrary prime number. (1)
(2)
is another arbitrary prime number. (3)
Example: ,