If $\frac{37}{13} = 2 + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}$, where $x$, $y$, $z$ are natural numbers, then $x$, $y$, $z$ are

### Concept & Strategy This problem requires converting an improper fraction into a **Continued Fraction**. The strategy is to repeatedly apply the Euclidean division algorithm: separate the integer part, and invert the remaining fractional part to find the next level's denominator. ### Step-by-Step Solution * **Step 1: Extract the first integer part** Divide 37 by 13. The quotient is 2, and the remainder is 11. $$\frac{37}{13} = 2 + \frac{11}{13}$$ Comparing this to the given expression, the "2" matches. We now need to format $\frac{11}{13}$ to match $\frac{1}{x + \dots}$. * **Step 2: Invert to find $x$** Rewrite $\frac{11}{13}$ as $\frac{1}{\frac{13}{11}}$. Now, extract the integer part of $\frac{13}{11}$: $$\frac{13}{11} = 1 + \frac{2}{11}$$ By comparing this to $\frac{1}{x + \dots}$, we see that **$x = 1$**. * **Step 3: Invert to find $y$ and $z$** Take the remaining fraction $\frac{2}{11}$ and invert it: $\frac{1}{\frac{11}{2}}$. Extract the integer part of $\frac{11}{2}$: $$\frac{11}{2} = 5 + \frac{1}{2}$$ By comparing this to $y + \frac{1}{z}$, we can clearly see that **$y = 5$** and **$z = 2$**. ### Exam Strategy & Shortcut Instead of writing out the full algebraic equations, just perform a chained division on your scratchpad. 1. $37 \div 13 = 2$ Remainder $11$. 2. Invert: $13 \div 11 = 1$ Remainder $2$. ($x=1$) 3. Invert: $11 \div 2 = 5$ Remainder $1$. ($y=5, z=2$) The sequence of quotients and the final divisor gives you $2, 1, 5, 2$. The unknowns $x, y, z$ map directly to $1, 5, 2$. ### Common Pitfall A common mistake is forgetting to invert the fraction at each step. Students might write $13/11$ and then incorrectly try to pull out a fraction from $11/13$ again, leading to loops or incorrect algebraic matching. Always remember the format requires $1$ in the numerator. ### Final Answer Therefore, the correct answer is **1, 5, 2**.