CAT 2024 Slot 3 QA Question 3: If $(a + b\sqrt{3})^{2} = 52 + 30\sqrt{3}$, where $a$ and $b$ are natural numbers, then $a + b$ equals

Question

If $(a + b\sqrt{3})^{2} = 52 + 30\sqrt{3}$, where $a$ and $b$ are natural numbers, then $a + b$ equals

Accepted Answer

Expand the left side and compare the rational and irrational parts.

$(a + b\sqrt{3})^{2} = a^{2} + 3b^2 + 2ab\sqrt{3}$
$a^2 + 3b^2 + 2ab\sqrt{3} = 52 + 30\sqrt{3}$
$2ab = 30$
$ab = 15$
$a^2 + 3b^2 = 52$

Among natural-number factor pairs of $15$, $(a,b) = (5,3)$ satisfies $a^2 + 3b^2 = 52$.

$a + b

If (a+b3)2=52+303(a + b\sqrt{3})^{2} = 52 + 30\sqrt{3}(a+b3​)2=52+303​, where aaa and bbb are natural numbers, then a+ba + ba+b equals