The smallest positive integer with exactly eight positive divisors is 24.
Its divisors are 1, 2, 3, 4, 6, 8, 12, and 24. The prime factorization explains the count: 24 = 2³ × 3. For a number written as prime powers, the number of positive divisors is found by adding 1 to each exponent and multiplying, giving (3 + 1)(1 + 1) = 8.
To see why it is the smallest, numbers below 24 cannot produce eight distinct divisors. The nearby candidates illustrate the issue: 18 has six divisors, 20 has six, and 22 has four. The next common eight-divisor example, 30, is larger than 24.
The phrase “positive divisors” matters. It includes 1 and the number itself, but not negative divisors or repeated factor expressions. Since 24 has more divisors than every smaller positive integer, it is also a highly composite number—the first number with that status after 12.