Absolute Value of Integer is not less than Divisors

Theorem

A (non-zero) integer is greater than or equal to its divisors in magnitude:

$\forall c \in \Z_{\ne 0}: a \divides c \implies a \le \size a \le \size c$

Corollary

Let $a, b \in \Z_{>0}$ be (strictly) positive integers.

Let $a \divides b$.

Then:

$a \le b$

Proof

Suppose $a \divides c$ for some $c \ne 0$.

From Negative of Absolute Value:

$a \le \size a$

Then:

\(\ds a\)

\(\divides\)

\(\ds c\)

\(\ds \leadsto \ \ \)

\(\ds \exists q \in \Z: \, \)

\(\ds c\)

\(=\)

\(\ds a q\)

Definition of Divisor of Integer

\(\ds \leadsto \ \ \)

\(\ds \size c\)

\(=\)

\(\ds \size a \size q\)

\(\ds \leadsto \ \ \)

\(\ds \size a \size q \ge \size a \times 1\)

\(=\)

\(\ds \size a\)

\(\ds \leadsto \ \ \)

\(\ds a \le \size a\)

\(\le\)

\(\ds \size c\)

$\blacksquare$

Also see

• Non-Zero Integer has Finite Number of Divisors: a direct corollary

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 11.2$: The division algorithm
• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Theorem $2 \text{-} 2 \ (6)$
• 1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Corollary $2$
• 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(iii)}$