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Home » Substitute \( p = 2 \) into (4): - High Altitude Science

Substitute \( p = 2 \) into (4): - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 \boxed{180}

📌 Question: What two-digit number is one more than a multiple of 9 and also one more than a multiple of 11?

📌 Solution: Let $ x \equiv 1 \pmod{9} $ and $ x \equiv 1 \pmod{11} $. This implies $ x \equiv 1 \pmod{99} $, since 9 and 11 are coprime. The two-digit numbers satisfying this are $ 99 + 1 = 100 $ (not two-digit) and $ 1 $. However, the next smaller solution is $ 1 $, which is not two-digit. Thus, no two-digit number satisfies both conditions.

📌 \boxed{1} (Note: The question may require reevaluation for valid constraints.)

📌 Question: What is the largest integer that must divide the product of any three consecutive integers in a population growth model?

📌 Solution: Among any three consecutive integers, there is at least one multiple of 2 and one multiple of 3. Thus, the product is divisible by $ 2 \times 3 = 6 $. For example, $ 1 \times 2 \times 3 = 6 $, $ 2 \times 3 \times 4 = 24 $, and $ 3 \times 4 \times 5 = 60 $, all divisible by 6. No higher integer (e.g., 12) divides all such products, as $ 1 \times 2 \times 3 = 6 $ is not divisible by 12.

📌 Question: What is the remainder when the sum of the first 4 prime numbers after 20 is divided by 5?

📌 Solution: The first 4 primes after 20 are 23, 29, 31, and 37. Their sum is $ 23 + 29 + 31 + 37 = 120 $. Dividing 120 by 5 gives a remainder of $ 120 - 5 \times 24 = 0 $.

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