Can you find the least possible number such that

If the number is divided by 3, it gives the remainder of 1;
If the number is divided by 4, it gives the remainder of 2;
If the number is divided by 5, it gives the remainder of 3;
If the number is divided by 6, it gives the remainder of 4.

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:

There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

When monkey rotate the gear, which mark will be hit 1 or 2 ?

There are people and strange monkeys on this island, and you can not tell who is who (Edit: until you understand what they said - see below). They speak either only the truth or only lies.
Who are the following two guys?
A: B is a lying monkey. I am human.
B: A is telling the truth.

Begin with a word, five letters to my name,
Remove the first and last but I am the same
Take out my middle and still, I remain.
What word am I?

Two prisoners Jack and Jill are locked in a cell.
There is open window approx 40 feet above the ground of the cell.
They are never able to reach there.

Then they plan to escape by a tunnel and they start digging out.
After digging for more than 20 days, Jill comes with the different plan and they escaped.

what was the plan ?

I look at you, you look at me, I raise my right, you raise your left. What am I?

I am lying and I always lie. Tell me whether I am lying or telling the truth.

What is the relationship between these matchstick pictures below?

I am a type of room you cannot enter or leave. Raised from the ground below, I could be poisonous or a delicious treat. What am I?