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Home » Surface area of a sphere = 4ÏrÂ² - Aurero

Surface area of a sphere = 4ÏrÂ² - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Thus, the remainder is $0$.

📌 Therefore, the remainder is $oxed{0}$.

📌 Question: What is the largest possible value of $\gcd(a,b)$ if the sum of two positive integers $a$ and $b$ is $2024$ and both are even?

📌 Solution: Let $a + b = 2024$, and suppose $d = \gcd(a,b)$. Since $d \mid a$ and $d \mid b$, then $d \mid (a + b) = 2024$. So $d$ is a divisor of $2024$.

📌 Additionally, since both $a$ and $b$ are even, $d$ must be even (because any common divisor of two even numbers is even).

📌 To maximize $d$, we take the largest even divisor of $2024$ such that $a$ and $b$ are positive integers.

📌 Factor $2024$:

📌 = 2^3 imes 11 imes 23.

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= 12^2 - 2ab = 144 - 2ab
Solve for \( ab \):
2ab = 144 - 80 = 64 \quad \Rightarrow \quad ab = 32
Now substitute into the formula for \( a^3 + b^3 \):
a^3 + b^3 = 1728 - 3(32)(12) = 1728 - 1152 = 576
\boxed{576}
A pharmacologist is modeling enzyme reaction rates with the equation \...
3a - 5 = 2(4 - a)
Expand the right-hand side:
3a - 5 = 8 - 2a

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