code: def lcim(numbers):
      """Returns the least common integral multiple of a list of numbers.
  
      The numbers can be rational or irrational or a mixture of both.
      `None` is returned for incommensurable numbers.
  
      Parameters
      ==========
  
      numbers : list
          Numbers (rational and/or irrational) for which lcim is to be found.
  
      Returns
      =======
  
      number
          lcim if it exists, otherwise ``None`` for incommensurable numbers.
  
      Examples
      ========
  
      >>> from sympy.calculus.util import lcim
      >>> from sympy import S, pi
      >>> lcim([S(1)/2, S(3)/4, S(5)/6])
      15/2
      >>> lcim([2*pi, 3*pi, pi, pi/2])
      6*pi
      >>> lcim([S(1), 2*pi])
      """
      result = None
      if all(num.is_irrational for num in numbers):
          factorized_nums = [num.factor() for num in numbers]
          factors_num = [num.as_coeff_Mul() for num in factorized_nums]
          term = factors_num[0][1]
          if all(factor == term for coeff, factor in factors_num):
              common_term = term
              coeffs = [coeff for coeff, factor in factors_num]
              result = lcm_list(coeffs) * common_term
  
      elif all(num.is_rational for num in numbers):
          result = lcm_list(numbers)
  
      else:
          pass
  
      return result
  
imports: from .accumulationbounds import AccumBounds, AccumulationBounds
  from .singularities import singularities
  from sympy.core import Pow, S
  from sympy.core.function import diff, expand_mul, Function
  from sympy.core.kind import NumberKind
  from sympy.core.mod import Mod
  from sympy.core.numbers import equal_valued
  from sympy.core.relational import Relational
  from sympy.core.symbol import Symbol, Dummy
  from sympy.core.sympify import _sympify
  from sympy.functions.elementary.complexes import Abs, im, re
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.integers import frac
  from sympy.functions.elementary.piecewise import Piecewise
  from sympy.functions.elementary.trigonometric import TrigonometricFunction, sin, cos, tan, cot, csc, sec, asin, acos, acot, atan, asec, acsc
  from sympy.functions.elementary.hyperbolic import sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh, acoth, asech, acsch
  from sympy.polys.polytools import degree, lcm_list
  from sympy.sets.sets import Interval, Intersection, FiniteSet, Union, Complement
  from sympy.sets.fancysets import ImageSet
  from sympy.sets.conditionset import ConditionSet
  from sympy.utilities import filldedent
  from sympy.utilities.iterables import iterable
label_desc: Use image node_id 6 for calling a global function with example usage: lcim(numbers) and returns: result