Consequences of the FTA

From the canonical form of a number several useful pieces of information can be read off immediately.

Example — Immediate consequences

From $n=p_1^{a_1}\cdots p_k^{a_k}$ one reads off at once:

• number of positive divisors: $\tau(n)=(a_1+1)(a_2+1)\cdots(a_k+1)$;
• $n$ is a perfect square $\iff$ all the $a_i$ are even;
• $a\mid b$ (”$a$ divides $b$”) $\iff$ every prime in $a$ appears in $b$ with exponent $\ge$.

For $168=2^3\cdot 3\cdot 7$: we have $\tau(168)=4\cdot 2\cdot 2=16$ divisors, and it is not a perfect square (the exponent $3$ is odd).

Links

Topics: Numbers and operations
Concepts: Divisibility · Number of divisors · Perfect square · Fundamental theorem of arithmetic
Skills: Factorise