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Home » eq d_2 $: 3 â 6 - Crosslake

eq d_2 $: 3 â 6 - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 $$ n = 16(5m + 1) = 80m + 16 $$

📌 The smallest positive $ n $ occurs when $ m = 0 $, giving $ n = 16 $. Verifying: $ 16 \div 16 = 1 $ and $ 16 - 1 = 15 $, which is divisible by 5.

📌 Question: What is the sum of all the squares of the divisors of $ 12 $?

📌 Solution: The divisors of $ 12 $ are $ 1, 2, 3, 4, 6, 12 $. Compute the sum of their squares:

📌 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2 = 1 + 4 + 9 + 16 + 36 + 144 = 210

📌 Question: What is the largest integer that must divide the product of any five consecutive integers in a neural network's orbital cycle?

📌 Solution: The product of $ k $ consecutive integers is divisible by $ k! $. For five consecutive integers, this is $ 5! = 120 $. For example, $ 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120 $, and $ 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 =

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Thus, the area of the larger square is \(\boxed{50}\) square cm.
An equilateral triangle has a perimeter of 18 cm. If each side is incr...
The side length \(s\) of the original equilateral triangle is:
s = \frac{18}{3} = 6 \text{ cm}
The area \(A_1\) of an equilateral triangle with side \(s\) is:
A_1 = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\...
If each side is increased by 2 cm, the new side length is \(8\) cm. Th...
A_2 = \frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 1...
A_2 - A_1 = 16\sqrt{3} - 9\sqrt{3} = 7\sqrt{3} \text{ square cm}
Thus, the area increases by \(\boxed{7\sqrt{3}}\) square cm.

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