There can be infinitely many distinct factors.
FactorsPerfect squares have an odd number of factors. If you think of factor pairs, the pair of factor that when multiplied together equals the given number, you realize that there are two factors for each pair until you come to the factor that when squared equals the number. For example, the factor pairs of 16 are 1 x 16, 2 x 8, and 4 x 4, giving us the factors of 1, 2, 4, 8, and 16 - an odd number of factors.Prime FactorsHowever, the situation is different with prime factors. It will depend on whether you list every prime factor or only the distinct prime factors. There will always be an even number of prime factors because when you square a number, you double the prime factors, which means you have an even number. That is not the case for distinct prime factors, because when you square a number, you are not adding any additional distinct factors. So, in some cases, there will be an even number of distinct prime factors and sometimes an odd number. Some examples follow.4 = 2 x 2. There are two prime factors, but one distinct prime factor.36 = 2 x 2 x 3 x 3. There are four prime factors and two distinct prime factors.900 = 2 x 2 x 3 x 3 x 5 x 5. There are six prime factors and three distinct prime factors.
Nine factors.
1 2 3 4 6 8 9 12 16 18 24 36 48 72 144.
The factors of 16 are 1,2,4,8,and 16The factors of 32 are 1,2,4,8,16, and 32Thus, the common factors of 16 and 32 are 1,2,4,8, and 16
1, 2, 4, 8, 16.
Distinct means separate. Sometimes, as in the case of square numbers, factors are repeated. The factor pairs of 16 are (16,1)(8,2)(4,4) The distinct factors of 16 are 1, 2, 4, 8, 16 The same is true of prime factors. The prime factorization of 36 is 2 x 2 x 3 x 3. The distinct prime factors of 36 are 2 and 3.
Five: 1, 2, 4, 8 and 16.
16=2*2*2*2 There is only one distinct prime factor, that is 2.
1, 2, 4, 8, 16, 32.
7 factors. They are: 1, 2, 4, 8, 16, 32, 64
1, 2, 4, 8, 16, 32, 64.
The distinct factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
Six distinct factors Two distinct prime factors
1 2 4 8 16 32 64 128.
Twelve distinct factors.
There can be infinitely many distinct factors.