​Letitia, Brianne, and Jake met at the mall on December 31. Letitia said that she intends to come to the mall every seventh day throughout the next year. Brianne said that she intends to be there...

​Letitia, Brianne, and Jake met at the mall on December 31. Letitia said that she intends to come to the mall every seventh day throughout the next year. Brianne said that she intends to be there every third day, and Jake said that he would be there every second day. Letitia said that she knew she would be at the mall a total of 52 days, since 365÷77 is 52 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 17 times, since they will both be coming every 21 days and 365÷21 is 17 with a remainder of 8.

a) How many times will all three friends meet at the mall in the next year?

b) Use reasoning similar to Letitia and Brianne to construct a three-loop Venn diagram that shows the number of times three friends go to the mall in all of the possible combinations.

c) How many days a year will Jake be at the mall by himself?

d) How many days in the year will none of the three be at the mall?