What number has 24 integers?

The number that has 24 integers as its factors is 60. This is because 60 can be factored into its prime components as (2^2 \times 3^1 \times 5^1). Using the formula for the number of divisors, ((e_1 + 1)(e_2 + 1)(e_3 + 1)) where (e_i) are the exponents of the prime factorization, we get ((2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 = 12). However, the number 840 also has 24 divisors, as it can be expressed as (2^3 \times 3^1 \times 5^1 \times 7^1), yielding ((3 + 1)(1 + 1)(1 + 1)(1 + 1) = 4 \times 2 \times 2 \times 2 = 32) which exceeds 24. Thus, numbers with exactly 24 divisors can include more than one value, like 360 or 720.