Shifting Times Tables

The numbers in the four times table are
$$4, 8, 12, 16...$$
I could shift the four times table up by 3 and end up with
$$7, 11, 15, 19...$$
What do you notice about the differences between numbers in each sequence?

The computer will display five numbers from a shifted times table.
On Levels 1 and 2 it will always be the first five numbers.
On Levels 3 and 4 it could be any five numbers.

Can you work out the computer's times tables and the shift that has taken place each time?

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Once you are confident that you can work out the times table and the shift quite easily, here are some questions to consider:

What can you say if the numbers are all odd?
What about if they are all even?
Or a mixture of odd and even?

What can you say if the units digits are all identical?
What if there are only two different units digits?

What can you say if the difference between two numbers is prime?
What can you say if the difference between two numbers is composite (not prime)?

Can you explain how you worked out the table and shift each time, and why your method will always work?