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==Prime factorization==

==Prime factorization==

The expression of an integer as a product of its prime factors is called a '''prime factorization'''.  The prime factorisation of 24 is

The expression of an integer as a product of its prime factors is called a '''prime factorization'''.  The prime factorization of 24 is

:<math>24 = 2 * 2 * 2 * 3 </math>

:<math>24 = 2 * 2 * 2 * 3 </math>

:<math>24 = 2^3 * 3</math>.

:<math>24 = 2^3 * 3</math>.

The [[Fundamental Theorem of Arithmetic|Prime Factorisation Theorem]] guarantees that every integer has a unique prime factorization, e.g. 24 =2³3¹, though it may have multiple non-prime factorizations (e.g. 24 = 2 * 12, 6 * 4, 3 * 8).

The [[Prime Factorization Theorem]] guarantees that every integer has a unique prime factorization, e.g. 24 =2³3¹, though it may have multiple non-prime factorizations (e.g. 24 = 2 * 12, 6 * 4, 3 * 8).

The number of divisors of an integer may be determined from its prime factorization when expressed in exponent form, by incrementing each exponent by 1 and multiplying the results.  In the example above, the exponents of prime factors 2 and 3 are 3 and 1, respectively.  The number of divisors of 24 is therefore

The number of divisors of an integer may be determined from its prime factorization when expressed in exponent form, by incrementing each exponent by 1 and multiplying the results.  In the example above, the exponents of prime factors 2 and 3 are 3 and 1, respectively.  The number of divisors of 24 is therefore