Letitia, Brianne, and Jake met at the mall on December 31 . Letitia said that she intends to come to the mall every fourth day throughout the next year, Brianne said that sh intends to be there every third day, and Jake said that he would be there every seventh day. Letitia said that she knew she would be at the mall a total of 91 days, since \( 365+4 \) is 91 with a remainder of 1 . Brianne said that she'd be at the mall 121 days, since \( 365+3 \) is 121 with a remainder of 2 . Moreover, Brianne said that of those 121 days, she would expect to see Letitia 30 times, since they will both be coming every 12 days and \( 365+12 \) is 30 with a remainder of 5 . Complete parts (a) through (d) belov (a) How many days will all three friends meet at the mall in the next year? There will be \( \square \) day(s) when all three meet.
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The Deep Dive
To determine how many days all three friends will meet at the mall, we need to find the least common multiple (LCM) of their visiting schedules: Letitia (every 4 days), Brianne (every 3 days), and Jake (every 7 days). The LCM of 4, 3, and 7 can be calculated as follows: - The prime factorization is: - 4 = \( 2^2 \) - 3 = \( 3^1 \) - 7 = \( 7^1 \) - The LCM will be formed by taking the highest powers of all prime factors: - LCM = \( 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84 \) This means they will all meet every 84 days. Now, we need to find how many times they meet within the year (365 days): 365 days ÷ 84 days = 4.35 So, they will meet 4 times in the year. Thus, there will be **4 days** when all three meet.
