How do we find divisors of a number? For example, how do we calculate the number of divisors of 900?

Answer: 900 = 2^{2} × 3^{2} × 5^{2}. Therefore, any number that is a factor of 900 can have powers of 2 equal to 2^{0}, 2^{1} or 2^{2}. Similarly, for 3, the powers can be 3^{0}, 3^{1}, or 3^{2} and for 5 they will be 5^{0}, 5^{1}, or 5^{2}. Writing the powers in a line we have-

Now any combination of a power of 2, a power of 3, and a power of 5 will give us a divisor. For example, in the figure, 2^{1} × 3^{2} × 5^{1} will be a divisor of 900. As we can select a power of 2 in 3 ways, a power of 3 in 3 ways, and a power of 5 in 3 ways, the total number of combinations will be 3 × 3 × 3 = 27. Therefore, the number of divisors of 900 is 27.