The sum of two numbers is 21 and their greatest common factor and least common multiple are 7 and 14 respectively. The sum of the reciprocals of two numbers is...

Steps to Solve

1. Identify the Variables Let the two numbers be $x$ and $y$.

Given:

• Their sum: $x + y = 21$
• The greatest common factor (GCF): $\text{GCF}(x, y) = 7$
• The least common multiple (LCM): $\text{LCM}(x, y) = 14$

2. Set Up Equations Using GCF and LCM We know that:

$$ \text{GCF}(x, y) \times \text{LCM}(x, y) = x \times y $$

Substituting the given values:

$$ 7 \times 14 = x \times y $$

This simplifies to:

$$ x \times y = 98 $$

3. Solve the System of Equations We now have a system of equations:
4. $x + y = 21$
5. $xy = 98$

From the first equation, we can express $y$ in terms of $x$:

$$ y = 21 - x $$

Substituting into the second equation:

$$ x(21 - x) = 98 $$

This simplifies to:

$$ 21x - x^2 = 98 $$

Rearranging gives:

$$ x^2 - 21x + 98 = 0 $$

4. Use the Quadratic Formula We can solve this quadratic equation using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Where $a = 1$, $b = -21$, and $c = 98$:

$$ x = \frac{21 \pm \sqrt{(-21)^2 - 4 \cdot 1 \cdot 98}}{2 \cdot 1} $$

Calculating the discriminant:

$$ (-21)^2 - 392 = 441 - 392 = 49 $$

Now substituting back into the quadratic formula:

$$ x = \frac{21 \pm 7}{2} $$

This gives us:

$$ x_1 = \frac{28}{2} = 14 $$

$$ x_2 = \frac{14}{2} = 7 $$

Thus $x$ and $y$ can be:

$$ (x, y) = (14, 7) \quad \text{or} \quad (7, 14) $$

5. Calculate the Sum of the Reciprocals Now, we can find the sum of the reciprocals:

$$ \frac{1}{x} + \frac{1}{y} = \frac{1}{14} + \frac{1}{7} $$

Finding a common denominator (14):

$$ \frac{1}{14} + \frac{2}{14} = \frac{3}{14} $$