Skip to content

Divisible

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Divisible[n, m]
    yields True if n is divisible by m, and False otherwise.
n is divisible by m when n is an integer multiple of m; this is
effectively Mod[n, m] == 0.  Works for machine and BigInt integers,
Gaussian integers, rationals, and exact numeric quantities (the
quotient n/m must reduce to an integer or Gaussian integer).  Returns
False unless n and m are manifestly divisible; symbolic, non-numeric
arguments are left unevaluated.  Listable.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Divisible[10, 2]
Out[1]= True

In[2]:= Divisible[5, 2]
Out[2]= False

In[3]:= Divisible[3 + I, 1 - I]
Out[3]= True

In[4]:= Divisible[2 Pi, Pi/2]
Out[4]= True

In[5]:= Divisible[Sqrt[6], Sqrt[2]]
Out[5]= False

In[6]:= Divisible[{1, 2, 3, 4, 5, 6}, 2]
Out[6]= {False, True, False, True, False, True}

Implementation notes

  • Machine integers and GMP bigints: tested directly with mpz_divisible_p, so large cases such as Divisible[10^3000 + 1, 16001] → True are exact. By the GMP convention, divisibility by 0 holds iff n == 0 (Divisible[0, 0] → True, Divisible[6, 0] → False); sign is ignored (Divisible[10, -2] → True).
  • Gaussian integers, rationals, and exact numeric quantities: the quotient n/m is formed and evaluated; the result is True iff it reduces to an integer or a Gaussian integer. So Divisible[3 + I, 1 - I] → True, Divisible[3/2, 1/2] → True, Divisible[2 Pi, Pi/2] → True, while Divisible[Sqrt[6], Sqrt[2]] → False.
  • Listable: threads element-wise over lists, e.g. Divisible[{1, 2, 3, 4, 5, 6}, 2] → {False, True, False, True, False, True}.
  • Symbolic, non-numeric arguments leave the call unevaluated (e.g. Divisible[x, 2]).
  • Diagnostics: too few arguments emit Divisible::argm, too many emit Divisible::argt; both leave the call unevaluated.

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References