Multiplication-Division-Remainders

http://https://nrich.maths.org/1783

I’m thinking of a number.
My number is both a multiple of 5 and a multiple of 6.
What could my number be?
What else could it be?
What is the smallest number it could be?
I’m thinking of a number.
My number is a multiple of 4, 5 and 6.
What could my number be?
What else could it be?
What is the smallest number it could be?

Here are some more questions you might like to consider:

I’m thinking of a number that is 1 more than a multiple of 7.
My friend is thinking of a number that is 1 more than a multiple of 4.
Could we be thinking of the same number?

I’m thinking of a number that is 3 more than a multiple of 5.
My friend is thinking of a number that is 8 more than a multiple of 10.
Could we be thinking of the same number?

I’m thinking of a number that is 3 more than a multiple of 6.
My friend is thinking of a number that is 2 more than a multiple of 4.
Could we be thinking of the same number?

Here’s a challenging extension:

We know that

Can you find a number with the property that when it is divided by each of the numbers 2 to 10, the remainder is always one less than the number it is has been divided by?
Can you find the smallest number that satisfies this condition?