8. . Jenny goes to karate class every 12 days and to the library every 18 days. On December 1 st she went to both class and the library. On what day will she do both??(the class and the library)

To determine when Jenny will go to both karate class and the library again, we need to find the least common multiple (LCM) of the two intervals: 12 days (karate class) and 18 days (library). ### Step 1: Find the prime factorization of each number. - The prime factorization of \(12\) is: \[ 12 = 2^2 \times 3^1 \] - The prime factorization of \(18\) is: \[ 18 = 2^1 \times 3^2 \] ### Step 2: Determine the LCM. To find the LCM, we take the highest power of each prime factor from the factorizations: - For \(2\), the highest power is \(2^2\). - For \(3\), the highest power is \(3^2\). Thus, the LCM is: \[ \text{LCM} = 2^2 \times 3^2 \] ### Step 3: Calculate the LCM. Now we calculate the LCM: \[ \text{LCM} = 4 \times 9 = 36 \] ### Step 4: Determine the date. Since Jenny went to both the karate class and the library on December 1st, we add 36 days to this date. Starting from December 1st: - December has 31 days, so from December 1st to December 31st is 30 days. - This means that 6 days into January will be the next occurrence. Thus, \(36\) days after December 1st is January 6th. ### Final Answer: Jenny will go to both karate class and the library again on **January 6th**.