数系与整除

Introduction: The Hierarchy of Numbers

Three Layers of Extension

Which of the following statements is correct?

Definition of Divisibility

Which divisibility relation holds?

Division with Remainder Theorem

What is the remainder of $31 \div 7$?

Greatest Common Divisor and Coprimality

$\gcd(12, 18) = ?$

Prime Numbers and the Fundamental Theorem of Arithmetic

Which of the following numbers is prime?

Example: Prime Factorization

What is the remainder of $100 \div 7$? Enter an integer.

$\gcd(24, 36) = ?$ Enter an integer.

Write $72$ as a product of prime powers and enter the exponent of $2$. ($72 = 2^a \cdot 3^b$, find $a$)

What is the equivalent characterization of $a \mid b$?

In your own words, explain: why extend $\mathbb{N}$ to $\mathbb{Z}$, and then $\mathbb{Z}$ to $\mathbb{Q}$? What problem does each extension solve?

Comprehensive Check