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📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Solution:** We want integers \( n \) such that \( 1 \leq n \leq 150 \) and \( n \equiv 3 \pmod{7} \).

📌 Such numbers are of the form \( n = 7k + 3 \).

📌 Find the smallest and largest such \( n \) in range:

📌 Smallest: \( k = 0 \Rightarrow n = 3 \)

📌 Largest: solve \( 7k + 3 \leq 150 \Rightarrow 7k \leq 147 \Rightarrow k \leq 21 \)

📌 So \( k = 0, 1, 2, \dots, 21 \) → total of \( 22 \) values.

📌 Thus, the number of such integers is \( \boxed{22} \).Question: What is the base-ten number represented by the base-eight number $ 321_8 $?

📌 Solution: To convert $ 321_8 $ to base ten, we expand it using powers of 8:

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