Homework 3
Facts About \(n_p\)
- If \(a\) and \(b\) are positive integers, then \(a = b\) if and only if \(n_p(a) = n_p(b)\) for all primes \(p\). (This is essentially the fundamental theorem of arithmetic.)
- \(n_p(ab) = n_p(a) + n_p(b)\) for all primes \(p\) and all positive integers \(a\) and \(b\). More generally,
\[
n_p(a_1\cdots a_n) \,=\, n_p(a_1) + \cdots + n_p(a_n)
\]
for all primes \(p\) and all positive integers \(a_1,\ldots,a_n\).
- If \(a\) and \(b\) are positive integers, then \(a\mid b\) if and only if \(n_p(a) \leq n_p(b)\) for all primes \(p\). In this case,
\[
n_p\biggl(\frac{b}{a}\biggr) \,=\, n_p(b) - n_p(a)
\]
for all primes \(p\).
- \(n_p\bigl(\mathrm{gcd}(a,b)\bigr) = \mathrm{min}\bigl(n_p(a),n_p(b)\bigr)\) for all primes \(p\) and all positive integers \(a\) and \(b\). More generally,
\[
n_p\bigl(\mathrm{gcd}(a_1\cdots a_n)\bigr) \,=\, \mathrm{min}\bigl(n_p(a_1),\ldots,n_p(a_n)\bigr)
\]
for all primes \(p\) and all positive integers \(a_1,\ldots,a_n\).
- \(n_p\bigl(\mathrm{lcm}(a,b)\bigr) = \mathrm{max}\bigl(n_p(a),n_p(b)\bigr)\) for all primes \(p\) and all positive integers \(a\) and \(b\). More generally,
\[
n_p\bigl(\mathrm{lcm}(a_1\cdots a_n)\bigr) \,=\, \mathrm{max}\bigl(n_p(a_1),\ldots,n_p(a_n)\bigr)
\]
for all primes \(p\) and all positive integers \(a_1,\ldots,a_n\).