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Answer:
96 ways
Step-by-step explanation:
Given
[tex]Day = 86400\ seconds[/tex]
Required
Ways to divide it into period of seconds
What this question implies is to determine the total number of factors of 86400
To start with, we determine the prime factorization of 86400
To do this, we continually divide 86400 by 2; when it can not be further divided, we divide by 3, then 7, then 11...
[tex]86400/2 = 43200[/tex]
[tex]43200/2=21600[/tex]
[tex]21600/2=10800[/tex]
[tex]10800/2=5400[/tex]
[tex]5400/2= 2700[/tex]
[tex]2700/2 = 1350[/tex]
[tex]1350/2=675[/tex]
[tex]675/3=225[/tex]
[tex]225/3=75[/tex]
[tex]75/3=25[/tex]
[tex]25/5=5[/tex]
[tex]5/5 = 1[/tex]
This implies that:
[tex]86400 = 2^7 * 3^3 * 5^2[/tex]
The number of factors d is the solved by:
[tex]d = (a+1)*(b+1) *(c+1)[/tex]
Where
[tex]n = 2^a * 3^b * 5^c[/tex]
By comparison:
[tex]a = 7[/tex]
[tex]b = 3[/tex]
[tex]c=2[/tex]
So:
[tex]d = (7+1)*(3+1) *(2+1)[/tex]
[tex]d = 8*4 *3[/tex]
[tex]d = 96[/tex]
Hence, there are 96 total ways