In how many ways can the number 6084 be written as a product of two different…

For any number N, the total factor count comes from adding 1 to each prime's exponent and multiplying those together. When N is a perfect square, exactly one of its factors equals the square root of N and pairs only with itself, while every other factor pairs with a distinct partner whose product with it equals N. So the number of ways to write N as a product of two DIFFERENT factors equals (total factor count minus 1) divided by 2.

1. Factorize 6084 into primes: 6084 = 22 x 32 x 132 (since 6084 = 4 x 9 x 169).

2. Total number of factors = (2+1)(2+1)(2+1) = 27.

3. Since 27 is odd, 6084 is a perfect square: the square root of 6084 is 78, and 78 x 78 = 6084.

4. Excluding this self-paired factor 78, the remaining 27 minus 1 = 26 factors combine into 26 divided by 2 = 13 pairs of two different factors whose product is 6084.

Independent check: listing the factor pairs of 6084 in order — (1, 6084), (2, 3042), (3, 2028), (4, 1521), (6, 1014), (9, 676), (12, 507), (13, 468), (18, 338), (26, 234), (36, 169), (39, 156), (52, 117) — gives exactly 13 pairs of different factors, plus the self-pair (78, 78), for 14 unordered pairs and 27 ordered factors in total, matching the count above.