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Mr. Prime always tell the truth when the day of the month is a prime number (i.e. the2 nd, 3rd, 5th, 7th,...), and lies on other days. Mr. Third always tell the truth on days that are multiplies of 3 (i.e. the 3rd, 6th, 9th, 12th,...), and lies on other days. On how many days in the month of January would both Mr. Prime and Mr. Third both say that they will lie the next day?

 Feb 24, 2021
 #1
avatar+943 
+2

First, find the days when they will tell the truth. January has 31 days:

 

Prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Third: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

 

The dates on which they will both tell the truth are the 3rd, so there is only date that works.

 Feb 24, 2021
 #2
avatar+30893 
+2

On how many days in the month of January would both Mr. Prime and Mr. Third both say that they will lie the next day?

Prime truth days    2  3   5   7   11   13   17   19    23    29    31 

Mr prime will say that he will lie the next day the 3rd (because he is telling the truth) and the 4th

1st  3rd  4th  5th    6th    7th   10th   11th  12th   13th   16th     17th   18th     19th    22nd    23rd   28th   29th  30th  31st

 

Mr third   

Third truth days   3 6 9 12 15 18 21 24 27 30      will say ' I will be lying tomorrow' on the following days

                    2nd  3rd      5th 6th     8th   9     11 12    14  15     17  18     20 21       23 24     26 27      29 30

 

I think !  Whew    !

Common days   3  5  6  11 12  17  18 23    29  30       TEN days

 Feb 25, 2021

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