How can you make number one disappear by adding something to it?

I hide my treasure in the ground, my tail is big and fluffy. If you spot me in a tree, please don’t call me scruffy. What am I?

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

I sleep during the day and fly at night, but I have no feathers to aid my flight. What am I?

I am the one who remains awake when you sleep. But I need to rest when you awake. Who Am I?

In an office's desk, you find a unique method that shows the current day of the month. Two numbered cubes are used to depict the current date.

Can you find out the numbers on both the cubes so that each date can be mentioned using them?

Please note that you have to use both the cubes to display any date. Thus the 1st day must be represented by 01 and so on.

You see a mathematical expression in the picture. Can you think of a movie that this expression refers to?

Chef Andrea has to light the kitchen stoves to open her restaurant for the day. There are three stoves in the kitchen: A glass stove, a brick stove, and a wood stove. She only has one match. Which does she light up first?

Place a mathematical symbol between the numerals 5 and 9 in such a way that the resulting number is greater than 5 but smaller than 9.

A number with an interesting property:

When I divide it by 2, the remainder is 1.
When I divide it by 3, the remainder is 2.
When I divide it by 4, the remainder is 3.
When I divide it by 5, the remainder is 4.
When I divide it by 6, the remainder is 5.
When I divide it by 7, the remainder is 6.
When I divide it by 8, the remainder is 7.
When I divide it by 9, the remainder is 8.
When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.
When I looked for a smaller number with this property I couldn't find one.

Can you find it?