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 3 × 7 = 21, and 21 + 2 = 23.

## The method

Add the following numbers, and of course, take mod 7, i.e. remove multiples of 7, whenever useful during the process of adding them.

### Explanation

The year-century-number

Add (century mod 4) to (year mod 4) and then take result × 5. Take that mod 7 if useful.

Note: when doing (year mod 4) get into the habit to check if it is a leap year! It is needed when dealing with January and February, see mnemonic table.

The year-number

Do (year mod 7) × 3.

The month-number

According to mnemonics, see table to the right.

The day-number

Just the day of the date as it is, but perhaps mod 7 if it helps.

Then take the result mod 7, and map that to the day of week. Sunday is 0 and so on.

## Example:

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

 January 6 Janis Joplin debuted with Big Brother in 1966 at the Avalon Ballroom.(1 less when leap year.) February 2 Fever (febrile) is often a 2ndary symptom.(1 less when leap year.) March 2 A march in 2/2 (time signature). April 5 The flower of the apricot has 5 petals. May 0 There are 0 bridges to Isle of May in Scotland. June 3 Juno had 3 children (Juventas, Mars, and Vulcan). July 5 In 55 BC Julius Caesar conducted the first Roman invasion of Britain. August 1 Augustus was the 1st emperor of the Roman Empire. September 4 Zeppo Marx was the 4th Marx brother October 6 An octopus who uses two of his arms to walk has 6 arms left to use as arms. November 2 A novel invention having 2 aspects, namely hardware and software - the computer. December 4 The Descendant is one of the 4 cardinal points in astrology (Ascendant, Descendant, Medium Coeli, and Imum Coeli).

# 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).