I also made a script where you can practice this method.
A method to calculate the day of week
Below I use the operator "mod" in the sense "take the remainder when dividing". For example, 23 mod 7 = 2, because 23 divided by 7 is 3 with remainder 2, since
Add the following numbers, and of course, take mod 7, i.e. remove multiples of 7, whenever useful during the process of adding them.
Then take the result mod 7, and map that to the day of week. Sunday is 0 and so on.
July 6, 2175.
The year-century-number is 6, because 21 mod 4 = 1, and 75 mod 4 = 3, adding 1 and 3 gives us 4, multiply that by 5 and we get 20, and if we do mod 7 we get 6.
The year-number is 15, because 75 mod 7 = 5, and 3 × 5 = 15. Add this to our 6 and we get 21, which gives 0 as remainder when divided by 7.
The month-number is 5 (because Julius Caesar invaded Britain in 55 BC), add that to our 0 and we have 5.
The day-number is 6. We add it to our 5 and get 11. We remove 7 from that, so we get 4.
So the day of the week is the 4th day, namely Thursday.
Mnemonics for the month numbers
Trick to find the remainder of division
The general idea
First find the next lower multiple of the divisor, and then check how many steps above your original number is.
For example, for finding the remainder of 97 divided by 4, we notice that the next lower multiple of 4 is 96, and since 97 is 1 more than that, the remainder is 1.
Specifics for division by 4
In our example of 97 above, the next lower multiple of 4 is 96. But how do we know it is 96 and not 94? The second digit obviously has to be even, but which one of them?
The trick you can use is like this: If the first digit is odd, like in the above case of 97, then the second digit must not be divisible by 4, whereas if the first digit is even then the second digit must be divisible by 4.
So, in our example, the next lower multiple of 4 must be 96, since 9 is odd and 6 is not divisible by 4.
If instead we have say 87, then the next lower multiple of 4 must be 84, because 8 is even, therefore we land at 4 since it is divisible by 4.
Just a note: 0 is even, and divisible by 4.
Specifics for division by 7
I don't know of any other method than remembering the multiplication table of 7 in combination with memorizing the few extra numbers up to 99.
So, apart from all the multiples up to 70, and of course 77 which is obvious, we have just three more, namely 84, 91 and 98 (which in a way correspond to 14, 21, and 28 above 70).