If a positive integer has 24 factors what is the least possible value of the positive integer?

Question

Simran Mehta · Accepted Answer

Prime Factorization and Number of Factors:If n has a prime factorization of the formn = p1^e1 * p2^e2 * ... * pk^ekwhere p1, p2, ..., pk are distinct primes and e1, e2, ..., ek are their respective powersthen the total number of factors of n is given by(e1 + 1)(e2 + 1) ... (ek + 1)We need to find the smallest n such that the product of its exponents plus one equals 24.
Finding Factor Combinations that Multiply to 24:The number 24 can be expressed as a product of integers in several ways:B6: 24 = 24 * 1B7: 24 = 12 * 2B8: 24 = 8 * 3B9: 24 = 6 * 4B10: 24 = 6 * 2 * 2B11: 24 = 4 * 3 * 2
Choosing the Smallest Combination:A13: To minimize n, we want to minimize the exponents and use the smallest primes possible.
Trying Different Combinations:For 24 = 8 * 3, the prime factorization would bep1^7 * p2^2 = 2^7 * 3^2 = 128 * 9 = 1152For 24 = 6 * 4, the prime factorization would bep1^5 * p2^3 = 2^5 * 3^3 = 32 * 27 = 864For 24 = 6 * 2 * 2, the prime factorization would be p1^5 * p2^1 * p3^1 = 2^5 * 3^1 * 5^1 = 32 * 3 * 5 = 480For 24 = 4 * 3 * 2, the prime factorization would bep1^3 * p2^2 * p3^1 = 2^3 * 3^2 * 5^1 = 8 * 9 * 5 = 360