What is the smallest positive integer n such that n is a multiple of 45, n is not divisibl… — GMAT

What is the smallest positive integer $n$ such that $n$ is a multiple of $45$ , $n$ is not divisible by $8$ , and $n$ has exactly $18$ positive integer divisors?

Answer: 180

Answer

180

The smallest positive integer satisfying all conditions is 180 because 180 = 2^2 * 3^2 * 5^1, which has (2+1)(2+1)(1+1) = 18 positive divisors, is a multiple of 45, and is not divisible by 8.

Step-by-Step Solution

Determine the prime factorization constraints for n

n must be of the form 2^a * 3^b * 5^c * ..., where b >= 2, c >= 1, and a <= 2.

n is a multiple of 45 = 3^2 * 5^1, requiring at least 3^2 and 5^1. Since n is not divisible by 8 = 2^3, the exponent of 2 cannot exceed 2.

Analyze the total number of divisors constraint

(a+1)(b+1)(c+1)... = 18

The total number of positive divisors is calculated by adding 1 to each exponent in the prime factorization and multiplying them together.

Find the optimal prime exponent configuration to minimize n

Factor 18 as 3 * 3 * 2, corresponding to exponents 2, 2, and 1.

Using three prime factors (2, 3, and 5) with smaller exponents yields a smaller total integer than using fewer prime factors with larger exponents.

Assign exponents to prime bases to yield the minimum integer value

n = 2^2 * 3^2 * 5^1 = 180

Assigning exponent 2 to base 2, exponent 2 to base 3, and exponent 1 to base 5 satisfies all constraints (b = 2 >= 2, c = 1 >= 1, a = 2 < 3) and minimizes the result.

Key Concept

Divisor count formula and prime factorization constraints

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