In how many ways the number 13260 be written as product of 2 factors

Let nn be any positive integer, and let d (n) d (n) denote the number of positive divisors of nn. Positive divisors of nn appear in pairs {a, na } {a, na }. Pairs of divisors aa, na na are distinct except when n = a2 n = a2. So if nn is not a perfect square, d (n) d (n) is even. If nn is a perfect square, then d (n) d (n) is odd. In other words,

d (n) d (n) is odd if and only if nn is a perfect square.

Determining one of aa, na na fixes the other ‘‘ ‘‘ complimentary "" divisor. Therefore the number of ways in which we can write n=a⋅b=a⋅ na n=a⋅b=a⋅ na is the number of ways in which we can choose aa.

If nn is not a perfect square, the number of such choices equals 12 d (n) 12 d (n).

If nn is a perfect square, the number of such choices equals 12 (d (n) - 1) 12 (d (n) - 1). We may combine the two cases by the expression

⌊ d (n) 2 ⌋ ⌊ d (n) 2 ⌋.