Question : The sum of the cubes of two numbers is 793. The sum of the numbers is 13. Then the difference between the two numbers is:
Option 1: 7
Option 2: 6
Option 3: 5
Option 4: 8
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Correct Answer: 5
Solution : Let the numbers be $a$ and $b$ where $a > b$. According to the question, $a^3+b^3=793$ and $a+b=13$ So, $(a+b)^3=a^3+b^3+3ab(a+b)$ ⇒ $13^3=793+3ab×13$ ⇒ $39ab=2197-793$ ⇒ $ab=\frac{1404}{39}$ $\therefore ab=36$ Now, $(a-b)^2=(a+b)^2-4ab$ ⇒ $(a-b)^2=13^2-4×36=169-144=25$ $\therefore (a-b)=\sqrt{25}=5$ Hence, the correct answer is $5$.
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