Question

how many factors are there in 1000?

Answer 1

To find the number of factors in a number,first compute the prime factorization of the number say \[N = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_n^{a_n } \]

From this form you can write that number of factors of the equation is \[(a_1+1) \times (a_2 +1) \times \cdots \times (a_n+1) \]

In this case \[ N = 1000 = 2^2 \times 5^2 \]

so the number of factors is \[3 \times 3 = 9 \]

Please note this formula could be proved in both algebraic and combinatorial ways,if you want that please mention.