Divisors Properties

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 \( N = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} \)

Let \(1=d_1<d_2<d_3 \cdots < d_{m-1}<d_m=N\) be the divisors of N.

\( \tau(r) = \{ x : x \mid r\} \). Example: \( \tau(p^\alpha) = \{1,p,p^2,p^3,\cdots,p^\alpha\}\), \(\tau(N) = \{ 1,d_2,d_3,\cdots,d_m\}\),\( |\tau(N)| = (\alpha_1+1)(\alpha_2+1)\cdots (\alpha_k+1)\)


  1. If \( a \mid N \) then \( \tau(a) \subseteq \tau(N) \). Also if \( a \mid d_i \) for some \(d_i \in \tau(N) \) then \( \tau(a) \subseteq \tau(N) \). I.e., All the divisors of \(a\) are also divisors of \(N\). Example: If \( 6 \mid N \implies \{1,2,3,6\}\subseteq \tau(N)\).

  2. \( d_i \times d_{m-i+1} = N\)

  3. If \( d_i \mid d_j \implies i < j \) i.e. \( j ≥ i+1\)

  4. If \(d_j \mid N\) and \( d_j > d_i\) then \( d_{i+1} ≤ d_j\).

  5. If \( \{ a,b \} \subset \tau(N) \implies \text{lcm}(a,b) \in \tau(N) \).

  6. \[m = |\tau(N)| = \left \{ \begin{matrix} \text{Prime} & \implies N = p^\alpha\\ \text{odd} & \iff N \text{is perfect square}\\ 2^\beta & \iff N \text{is squarefree}\\ \cdots & \cdots \end{matrix} \right. \]
  7. If \(2 \mid N\)(or \( 2k \mid N \) for some \( k \in \mathbb{N}\) ) then \( d_2 = 2\). If \( d_2 \ne 2 \implies 2 \nmid d_i \) where \( d_i \in \tau(N)\) (i.e. all divisors are odd in that case).
  8. If \( 3 \mid N\) (or \( 3k \mid N\) for some \( k \in \mathbb{N} \) ) then \( d_2 = 3\) if \( 2 \nmid N\) else \( d_3 =3\).
  9. If \( \{2,3\} \subset \tau(N)\) then \( d_1 = 1, d_2 = 2, d_3 = 3\) \[d_4 =  \left \{\begin{matrix} 4 & \text{if } 4 \mid N \\  5 & \text{if } 4 \nmid N \text{ and } 5\mid N\\  6 & \text{if } 4 \nmid N \text{ and } 5\nmid N \end{matrix}  \right. \]

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