The greatest power of 36 that will divide into 90! can be obtained by the formula :
Highest power of 2^n in 90!/2^n
Hence the number 21 as the exponent of 36 will divide the 90! with no remainder.
Prime factors of 36 are
36= 2×2×3×3= 2²×3²
In 90! the number of factors of 3 is 30+10+3+1=44
[tex]\frac{90}{3} + \frac{90}{9} +\frac{90}{27} +\frac{90}{81}[/tex]
= 30+ 10+ 3+1= 44
In 90! the number of factors of 2 is 45+22+11+ 5+2+1= 86
[tex]\frac{90}{2} + \frac{90}{4} + \frac{90}{8} +\frac{90}{16} + \frac{90}{32} +\frac{90}{64}[/tex]
=45+22+ 11+ 5+ 2+1= 86
In 90! the number of factors of 5 is 18+3=21
[tex]\frac{90}{5} +\frac{90}{25}[/tex]
= 18+ 3=21
The number of prime factors of 2 in 90! is more than 4 times the number of prime factors of 5.
The number of prime factors of 3 in 90! is more than 2 times the number of prime factors of 5.
This shows the number of prime factors of 5 is what limits the power of 36 that divided into 90!
Hence the number 21 as the exponent of 36 will divide the 90! with no remainder.
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