#### Answer

30 days later, or July 1st.

#### Work Step by Step

You and your brother both work the 4 until midnight shift. You have every 6th day off and your brother has every 10th day off. If you are both off on June 1, when will you both be off on the same day again?
To find the number of days until you are both off again, we need to find the Least Common Multiple (LCM) of 6 and 10.
We can do this by prime factoring and taking the factors needed to make up each number.
6 = 2 x 3
10 = 2 x 5
To get the LCM, we need a 2, because there is only one 2 in each factorization. We also need to have a 3, since the 6 has a factor of 3 in it and a 5 since the 10 has a factor of 5 in it.
2 x 3 x 5 = 30
You will both be off on the same day in 30 days. If you are off June 1, 30 days from that is July 1.