For each positive integer \(n\), \(\text{LCM}(1,2,\ldots,n)\) is the least common multiple of \(1, 2, \ldots, n.\) That is, the smallest positive integer divisible by each of \(1, 2, \ldots, n.\)
Determine all positive integers \(n\), with \(1\leq n \leq 100\) such that \[\text{LCM}(1, 2, \ldots, n) = \text{LCM}(1, 2, \ldots, n+4)\]
Note: In solving this problem, it might be helpful to know that we can calculate the LCM of a set of positive integers by
determining the prime factorization of each integer in the set,
determining the list of prime numbers that occur in these prime factorizations,
determining the highest power of each prime number from this list that occurs in the prime factorizations, and
multiplying these highest powers together.
For example, \(\text{LCM}(1, 2, 3, 4, 5, 6, 7, 8) = 2^3\cdot 3^1 \cdot 5^1 \cdot 7^1 = 840\), since the prime factorizations of \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), and \(8\) are \(2\), \(3\), \(2^2\), \(5\), \(2\cdot 3\), \(7\), and \(2^3\), respectively.