To solve this problem, we need to define the two numbers and create equations based on the given information. Let's call the two numbers \(\( a \)\) and \(\( b \)\), where \(\( a \)\) is the larger number.
Given:
1. The product of two numbers is 572.
\(\[ a \cdot b = 572 \]\)
2. The sum of the numbers is 12 times their positive difference.
\(\[ a + b = 12 \cdot |a - b| \]\)
We also know that since \(\( a \)\) is the larger number, the absolute value is not necessary since \(\( a - b \)\) will always be positive. Therefore, we can rewrite the second equation as:
\(\[ a + b = 12(a - b) \]\)
Now let's solve the equations.
Equation 1:
\(\[ a \cdot b = 572 \]\)
\(\[ b = \frac{572}{a} \]\)
Equation 2:
\(\[ a + b = 12(a - b) \]\)
\(\[ a + \frac{572}{a} = 12a - 12\frac{572}{a} \]\)
To eliminate the fraction, multiply through by \(\( a \)\):
\(\[ a^2 + 572 = 12a^2 - 6876 \]\)
Reorder terms and combine like terms:
\(\[ 11a^2 - a^2 = 6876 + 572 \]\)
\(\[ 11a^2 - a^2 = 7448 \]\)
\(\[ 10a^2 = 7448 \]\)
\(\[ a^2 = \frac{7448}{10} \]\)
\(\[ a^2 = 744.8 \]\)
Since \(\( a \)\) is a real number, we take the square root of both sides:
\(\[ a = \pm\sqrt{744.8} \]\)
However, we are looking for the larger number which must be positive:
\(\[ a = \sqrt{744.8} \]\)
The square root is slightly below \(\( \sqrt{729} = 27 \)\) and we are looking for integers since the product of the numbers is an integer and the sum also suggests they should be integers. Therefore, the closest integer is \(\( a = 26 \)\). But we must verify if \(\( b \)\), the smaller number, is also an integer.
\(\[ b = \frac{572}{a} \]\)
\(\[ b = \frac{572}{26} \]\)
\(\[ b = 22 \]\)
So the two numbers that satisfy both conditions are 26 and 22. Since \(\( a \)\) is the larger number:
The larger of the two numbers is 26.
