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IntegerExponent

Status: Stable

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

Description

IntegerExponent[n, b] gives the highest power of b that divides n.
IntegerExponent[n] is equivalent to IntegerExponent[n, 10] and gives the number of trailing zeros in the decimal digits of n.
IntegerExponent ignores the sign of n; IntegerExponent[0, b] is Infinity.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= IntegerExponent[1230000]
Out[1]= 4

In[2]:= IntegerExponent[2^10 + 2^7, 2]
Out[2]= 7

In[3]:= IntegerExponent[144, 2]
Out[3]= 4

In[4]:= IntegerExponent[100!, 2]
Out[4]= 97

In[5]:= IntegerExponent[0]
Out[5]= Infinity

In[1]:= IntegerExponent[]
Out[1]= IntegerExponent[]

In[2]:= IntegerExponent[1, 2, 3, 4]
Out[2]= IntegerExponent[1, 2, 3, 4]

In[1]:= IntegerExponent[1.123]
Out[1]= IntegerExponent[1.123]

Implementation notes

builtin_integerexponent returns the largest k with b^k | n (the base-b valuation, default base 10 = trailing-zero count). For base 2 it uses mpz_scan1 (position of the lowest set bit = 2-adic valuation); otherwise GMP's mpz_remove(q, |n|, base), which divides out base repeatedly and returns the count in one library call (intexp_count). IntegerExponent[0, b] is Infinity (every power divides 0). Validates arity (::argt), numeric non-integer n (::int), and base >= 2 (::ibase); symbolic n returns NULL.

  • Protected, Listable. Threads element-wise over a list of integers in

Attributes: Listable, Protected.

Implementation status

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

References

Notes & additional examples

Worked examples

In[1]:= IntegerExponent[1000]
Out[1]= 3

In[1]:= IntegerExponent[1000, 2]
Out[1]= 3

In[1]:= IntegerExponent[100!, 5]
Out[1]= 24

In[1]:= IntegerExponent[20!, 2]
Out[1]= 18

Notes

IntegerExponent[n, b] gives the highest power of b dividing n; with no base it counts trailing decimal zeros. The factorial examples reproduce Legendre's formula: 100! ends in IntegerExponent[100!, 5] = 24 zeros (the number of factors of five), and 20! contains 2^18. The sign of n is ignored and IntegerExponent[0, b] is Infinity.