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A Mala is a string of beads used to count mantras (Sanskrit prayers) in sets of 108 repetitions as a form of meditation.

Some of the most interesting are:

Astrology. There are 12 constellations and 9 arc segments. There are 12 houses and 9 planets. 12 X 9 = 108

Sanskrit alphabet has 54 letters. Each letter has a masculine (Shiva) and feminine (Shakti) energy 54 X 2 = 108

Heart Chakra: The chakras are the intersections of energy lines, and there are said to be a total of 108 energy lines converging to form the heart chakra. One of them, sushumna leads to the crown chakra, and is said to be the path to Self-realization

Sun and Earth: The diameter of the Sun is 108 times the diameter of the Earth.

The distance from the Sun to the Earth is 108 times the diameter of the Sun

Moon and Earth: The average distance of the Moon from the Earth is 108 times the diameter of the Moon.

In Ayurveda, there are 108 "Marm" points that are vital
for giving life to living beings.

Powers of 1, 2, and 3 in math: 1 to 1st power=1; 2 to 
2nd power=4 (2x2); 3 to 
3rd power=27 (3x3x3). 1x4x27=108

In Tantra, it is estimated that every day we breathe
21,600 times out of which 10,800 are solar energy and
10, 800 are lunar energy. Multiplying 108 X 100 is
10,800.

The famous sage Bharat wrote "Natya Shastra"which has 108 karanas (108 poses of Tandav by movement of hand & feet)

Harshad number: 108 is a Harshad no.which is an integer divisible by the sum of its digits (Harshad is from Sanskrit and means "great joy")

River Ganga: The sacred River Ganga spans a longitude of 12 degrees (79 to 91), & a latitude of 9 degrees (22 to 31). 12 times 9 equals 108.

Sri Yantra:

On the Sri Yantra there are marmas where three lines
intersect,& there are 54 such intersections.Each intersection has masculine and feminine, shiva and shakti qualities

54 times 2= 108. Thus, there are
108 points that define the Sri Yantra as well as the human body

1, 0, and 8: Some say that 
1 stands for God or higher Truth, 
0 stands for emptiness or completeness in spiritual practice, and 8 stands for infinity or eternity

Each zodiac sign has a total of 9 padas. There are 12 zodiac signs. So, 12*9 = 108 padas.

Pranayama: If one is able to be so calm in meditation as to have only 108 breaths in a day, enlightenment will come

Still You will find many reasons behind it, hence 108 signifies the wholeness of the divinity,perfect totality. So, let us follow what our ancestors told us to do.

Source: Twitter (Vशुद्धि @V_Shuddhi)