Sia

{'context': "[{'Student': 'my name is 4538291178, and I am currently in College', 'Sia': 'nice to meet you!'}, {'Student': '\\nFreeze on\\n[SMC 2012 Q22] Which of the following numbers does not have a square root in the form $x+y \\\\sqrt{3}$, where $x$ and $y$ are positive integers?\\n$19+15 \\\\sqrt{3}$\\n$28+15 \\\\sqrt{3}$\\n$45+15 \\\\sqrt{3}$\\n$65+15 \\\\sqrt{3}$\\n$82+15 \\\\sqrt{3}$', 'Sia': ' step 1 ⋮ To determine which number does not have a square root in the form $x + y\\\\sqrt{3}$, we start by expressing the square of $x + y\\\\sqrt{3}$: \\\ (x + y\\\\sqrt{3})^2 = x^2 + 2xy\\\\sqrt{3} + 3y^2 \\\\. This implies that for a number of the form $a + b\\\\sqrt{3}$, we need to equate both the integer and the coefficient of $\\\\sqrt{3}$.   step 2 ⋮ We can set up the respective equations from the integer and irrational parts.  ...  \\\\. Checking \\\\( x^2 + 3y^2 = 82 \\\\) does not yield valid positive integer solutions. answer choice:E'}, {'Student': 'What requirements need to be satisfied for a number to have a square root in the form $x + y\\\\sqrt{3}$, where $x$ and $y$ are positive integers?', 'Sia': ' step 1 ⋮ To determine the requirements for a number to have a square root in the form $x + y\\\\sqrt{3}$, we start by writing the square of $x + y\\\\sqrt{3}$: \\\\( (x + y\\\\sqrt{3})^2 = x^2 + 2xy\\\\sqrt{3} + 3y^2 \\\\.   step 2 ⋮ Equating the integer part and the coefficient of $\\\\sqrt{3}$ gives us necessary conditions that the integer must conform to regarding even and odd coefficients.'}]", 'question': ''}