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The Bully Mnemonic is a technique for remembering the exact (eight digits) number of seconds that occur in Earth's sidereal year and tropical year; a good approximation (four digits) of the Earth's Great Year; and a rough approximation of the Solar System's galactic year.

The following relationships are encoded in the Bully Mnemonic:

$${\displaystyle {1\,Sidereal\,Year}={31,558,150\,Seconds}}$$

$${\displaystyle {1\,Tropical\,Year}={31,556,926\,Seconds}}$$

$${\displaystyle 1\,Great\,Year\approx 25,824\,Sidereal\,Years\approx 25,825\,Tropical\,Years}$$

$${\displaystyle {1\,Galactic\,Year}\approx 213,417,800\,Tropical\,Years}$$

The first step is to write down the first five digits:

$${\displaystyle {\begin{matrix}1&2&3&4&5\end{matrix}}}$$

The second step is to select odd digits and intersperse them with zeros to form integers a) and b) as shown below: (important to remember that the first integer ends with 33 followed by a 0, whereas the second integer ends with 55 with no trailing 0)

$${\displaystyle {\begin{matrix}{\color {Red}1}&\scriptstyle {\text{2}}&{\color {Red}3}&\scriptstyle {\text{4}}&{\color {Red}5}\end{matrix}}}$$

$${\displaystyle a)\,{\color {Red}1}0{\color {Red}33}0}$$ $${\displaystyle b)\,{\color {Red}3}0{\color {Red}55}}$$

The third step is to select even digits and define numbers c) and d) as shown below:

$${\displaystyle {\begin{matrix}\scriptstyle {\text{1}}&{\color {Red}2}&\scriptstyle {\text{3}}&{\color {Red}4}&\scriptstyle {\text{5}}\end{matrix}}}$$

$${\displaystyle c)\,{\color {Red}2}}$$ $${\displaystyle d)\,0.{\color {Red}4}0}$$

Multiply integers a) and b) from Step 2 to get the total number of seconds in a sidereal year.

$${\displaystyle {\color {Red}1}0{\color {Red}33}0\times {\color {Red}3}0{\color {Red}55}=31558150={\frac {1\,Sidereal\,Year}{1\,Second}}}$$

Using Long Multiplication:

       3055
×     10330
————————————
       0000
      9165
     9165
    0000
   3055
————————————
   31558150

The tropical year has a slightly shorter duration than the sidereal year. The exact number of seconds in a tropical year is obtained by reducing integer a) by amount d), multiplying by b), and then reducing by c).

$${\displaystyle (({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)\times {\color {Red}3}0{\color {Red}55})-{\color {Red}2}=31556926={\frac {1\,Tropical\,Year}{1\,Second}}}$$

Using the Distributive Property of Multiplication:

(10330 - 0.40) × 3055 = (10330 × 3055) - (0.40 × 3055)
                      =    31558150    -     1222
                      =    31556928

The Great Year is, by definition, a least common multiple of the sidereal year and the tropical year. From steps 4 and 5 above, we have that the ratio of tropical years to sidereal years is:

$${\displaystyle {\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {(10330-0.40)\times 3055\,sec}{10330\times 3055\,sec}}}$$

Divide top and bottom by amount d) and use the Distributive Property of Multiplication to obtain:

$${\displaystyle {\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {({\frac {10330}{0.40}}-{\frac {0.40}{0.40}})\times 3055\,sec}{({\frac {10330}{0.40}})\times 3055\,sec}}}$$

From whence:

$${\displaystyle {\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {(25825-1)\times 3055\,sec}{(25825)\times 3055\,sec}}}$$

Consequently:

$${\displaystyle {\frac {25825\,Tropical\,Year}{25824\,Sidereal\,Year}}\approx {\frac {25825\times (25824)\times 3055\,sec}{25824\times (25825)\times 3055\,sec}}=1}$$

Finally:

$${\displaystyle 1\,Great\,Year\approx 25825\,Tropical\,Years\approx 25824\,Sidereal\,Years}$$

In terms of Long Multiplication; 0.40, 25825, and 10330 are related as follows:

       0.40
×  25825
————————————
       2.00
      08.0
     320
    200
   080
————————————
   10330.00

Multiply integer c) by the square of integer a) to get a rough approximate galactic year (the number of tropical years required for the Solar System to orbit once around the galactic center).

$${\displaystyle {\color {Red}2}\times {{\color {Red}1}0{\color {Red}33}0}^{2}=213417800\approx {\frac {1\,Galactic\,Year}{1\,Tropical\,Year}}}$$

Using Long Multiplication:

       10330
×      10330
——————————————
       00000
      30990
     30990
    00000
   10330
——————————————
   106708900

And finally:

106708900 × 2 = 213417800