To express the repeating decimal \( 3.765765765\ldots \) as a rational number, we can let \( x = 3.765765765\ldots \).
First, we can break this down as follows: \[ x = 3 + 0.765765765\ldots \]
Next, focus on the repeating part \( 0.765765765\ldots \). We can let: \[ y = 0.765765765\ldots \]
To eliminate the repeating part, we multiply \( y \) by \( 1000 \) (since the repeat length is 3 digits): \[ 1000y = 765.765765765\ldots \]
Now, we can set up the equation: \[ 1000y = 765 + y \] Subtract \( y \) from both sides: \[ 999y = 765 \]
Now, solve for \( y \): \[ y = \frac{765}{999} \]
Next, we need to simplify \( \frac{765}{999} \). We can find the greatest common divisor (gcd) of 765 and 999.
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Finding the prime factorizations:
- \( 765 = 3^2 \times 5 \times 17 \)
- \( 999 = 3^3 \times 37 \)
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Finding the gcd: The common factor is \( 3^2 \), so: \[ \text{gcd}(765, 999) = 9 \]
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Dividing both numerator and denominator by the gcd: \[ \frac{765 \div 9}{999 \div 9} = \frac{85}{111} \]
Now we have \( y = 0.765765765\ldots = \frac{85}{111} \).
Returning to our original \( x \): \[ x = 3 + y = 3 + \frac{85}{111} \]
We can express \( 3 \) as a fraction with a denominator of 111: \[ 3 = \frac{333}{111} \]
Adding these fractions gives: \[ x = \frac{333}{111} + \frac{85}{111} = \frac{333 + 85}{111} = \frac{418}{111} \]
Finally, we need to ensure \( \frac{418}{111} \) is in simplest form. The prime factorization of 418 is \( 2 \times 209 \) (and 209 is prime), and 111 is \( 3 \times 37 \). There are no common factors between 418 and 111, so the fraction is already in simplest form.
Thus, the rational representation of \( 3.765765765\ldots \) is: \[ \boxed{\frac{418}{111}} \]