Is Q a Perfect Square?

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Introduction

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Is Q a Perfect Square?

Introduction

In mathematics, a perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 4^2. Not all numbers are perfect squares, however. For example, 10 is not a perfect square because it cannot be expressed as the square of an integer.

In this article, we will explore the question of whether or not Q is a perfect square. Q is a rational number that is defined as follows:

Q = (1 + √5) / 2

We will use a variety of mathematical techniques to determine whether or not Q is a perfect square.

Method 1: Direct Calculation

One way to determine whether or not Q is a perfect square is to simply calculate its square root. If the square root is an integer, then Q is a perfect square. If the square root is not an integer, then Q is not a perfect square.

To calculate the square root of Q, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = 1, and c = -5. Substituting these values into the quadratic formula, we get:

x = (-1 ± √(1^2 - 4(1)(-5))) / 2(1)

x = (-1 ± √21) / 2

Therefore, the square root of Q is (-1 ± √21) / 2. Since this is not an integer, Q is not a perfect square.

Method 2: Prime Factorization

Another way to determine whether or not Q is a perfect square is to prime factorize it. If Q can be expressed as the product of two equal prime numbers, then Q is a perfect square. If Q cannot be expressed as the product of two equal prime numbers, then Q is not a perfect square.

To prime factorize Q, we can use the following steps:

1. Divide Q by the smallest prime number that divides it, which is 5.
2. Repeat step 1 until Q is no longer divisible by any prime number.

When we prime factorize Q, we get:

Q = (1 + √5) / 2

Q = (1 + 5^1/2) / 2

Q = 5^1/2 / 2

Therefore, Q cannot be expressed as the product of two equal prime numbers. Thus, Q is not a perfect square.

Method 3: Congruence Modulo 4

A third way to determine whether or not Q is a perfect square is to use congruence modulo 4. This method is based on the following theorem:

Theorem: A number is a perfect square if and only if it is congruent to 0 or 1 modulo 4.

To use this theorem, we can simply calculate Q modulo 4. We get:

Q ≡ (1 + √5) / 2 (mod 4)

Q ≡ 1 (mod 4)

Since Q is congruent to 1 modulo 4, it is not a perfect square.

Conclusion

We have used three different methods to determine whether or not Q is a perfect square. All three methods have shown that Q is not a perfect square. Therefore, we can conclude that Q is not a perfect square.

Closure

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