Divisor Count and the Missing Exponent Pattern
In a number-theory mock, the problem asks for the least positive integer N with exactly 12 positive divisors, divisible by 6, and not divisible by 9. A candidate writes that since 12=3·2·2, the exponent multiset in N=2^a3^b5^c... must be {2,1,1}. To satisfy divisibility by 6 but not 9, they put exponent 1 on 3, exponent 2 on 2, and exponent 1 on 5, obtaining N=2^2·3·5=60. They submit 60 and assert that any other assignment of primes or exponents gives a larger value. The candidate does not discuss the factorisations 12=4·3, 12=6·2, or 12=12, nor whether a prime larger than 5 could appear. The answer key records 60, so the numerical answer happens to agree, but the justification is incomplete.
1. Does τ(N)=12 force exponent pattern {2,1,1}? List all possible exponent patterns and explain which can occur under the divisibility conditions.
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No. If N=∏p_i^{e_i}, then τ(N)=∏(e_i+1)=12. The possible exponent multisets are {11}, {5,1}, {3,2}, and {2,1,1}. Because 6|N, the exponent of 3 is at least 1; because 9∤N, it is at most 1, hence exactly 1. So any valid pattern must contain an exponent 1. This immediately rules out {11} and {3,2}, since neither contains a 1. Pattern {5,1} would require 3^1 times another prime to the fifth power; the smallest is 2^5·3=96>60. Pattern {2,1,1} can use 3^1 and 2^2, with the remaining exponent 1 on the smallest unused prime 5, giving 60. Thus {2,1,1} is the only pattern capable of yielding a number below 96, but the candidate did not justify excluding the others.
2. Give a complete minimality proof for N=60.
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Any valid N has τ=12 and 3-adic exponent exactly 1. From the previous part, patterns with no exponent 1 are impossible; the {5,1} pattern has minimum 2^5·3=96. In pattern {2,1,1}, 3 must be one of the exponent-1 primes. To minimize, assign the remaining exponent 1 to 5 and exponent 2 to 2, since using a larger prime in place of 2 or 5 only increases N. This gives 2^2·3·5=60. It satisfies 6|60, 9∤60, and τ(60)=(2+1)(1+1)(1+1)=12. Since every other valid N is either at least 96 or larger than 60 within the same pattern, 60 is least.