Final answer:
Jan is thinking of the number 360, which has exactly 16 positive divisors, including 12 and 15. This was found by determining the least common multiple of 12 and 15 and finding a prime factorization to yield 16 divisors.
Step-by-step explanation:
Jan is thinking of a positive integer that has exactly 16 positive divisors, including 12 and 15. To find this number, let's begin by prime factorizing 12 and 15, yielding 12 = 2² × 3 and 15 = 3 × 5. Since the integer in question has both 12 and 15 as divisors, it must be a multiple of both, which also implies it is a multiple of their least common multiple (LCM), which is 2² × 3 × 5 or 60. This tells us the unknown integer will have at least the prime factors 2, 3, and 5 included in its prime factorization with certain exponents.
The number of divisors of an integer is determined based on the exponents in its prime factorization. If the prime factorization is a⁺ × b⁻ × c⁼…, then the number of divisors is (a+1) × (b+1) × (c+1)…. Our task is to find a set of exponents for the prime factors 2, 3, and 5 (and possibly others) such that the product of one more than each exponent equals 16.
One possible set of exponents for the prime factors that would give us 16 divisors is 2³ × 3² × 5¹, because (3+1)(2+1)(1+1) equals 16. Hence, Jan's number could be 2³ × 3² × 5¹ or 8 × 9 × 5, which equals 360. Therefore, the positive integer Jan is thinking of that has exactly 16 positive divisors is 360.