Find the highest power of 16 in 100!
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Concept:
Power is defined by a bse number and sn exponent. The base number represents the number that is being multiplied to get the power.
Given:
[tex]16[/tex] in [tex]100![/tex]
Find:
The highest power of [tex]16[/tex] in [tex]100![/tex]
Solution:
According to the problem,
[tex]16=2*8[/tex]
This can be simplied more into,
[tex]16=2*2*4[/tex]
Using floor function, the highest power of [tex]2[/tex] in [tex]100![/tex] is
[tex]\frac{100}{2}+\frac{100}{2^2} +\frac{100}{2^3} +\frac{100}{2^4} +\frac{100}{2^5} +\frac{100}{2^6} +\frac{100}{2^7}+...........[/tex]
[tex]50+25+12+6+3+1+0=97[/tex]
Using floor function, the highest power of [tex]4[/tex] in [tex]100![/tex] is
[tex]\frac{100}{4}+\frac{100}{4^2} +\frac{100}{4^3} +...........[/tex]
[tex]25+6+1=32[/tex]
Since [tex]97[/tex] is higher exponent thus,
[tex]2^{32}*2^{32}*4^{32}[/tex]
[tex]=16^{32}[/tex] is the highest power
Hence, the highest power of [tex]16[/tex] in [tex]100![/tex] is [tex]16^{32}[/tex]