Find the index of the greatest power (IGP) of 24 contained in 360!
Answers
Given : 360!
To Find : index of the greatest power (IGP) of 24 contained in
Solution:
24 = 2 x 2 x 2 x 3
=> 24 = 2³ x 3
Number of factors of 2
= [360/2¹] + [360/2²] + [360/2³ ] + [360/2⁴ ] + [360/2⁵ ] + [360/2⁶ ] + [360/2⁷ ] + [360/2⁸ ] ( 2⁹ > 360 )
= [360/2] + [360/4] + [360/8 ] + [360/16 ] + [360/32 ] + [360/64 ] + [360/128 ] + [360/256 ]
= 180 + 90 + 45 + 22 + 11 + 5 + 2 + 1
= 356
as 2 has power of 3
[356/3] = 118
Number of factors of 3
= [360/3¹] + [360/3²] + [360/3³ ] + [360/3⁴ ] + [360/3⁵ ] (3⁶ >360)
= [360/3] + [360/9] + [360/27 ] + [360/81 ] + [360/243 ]
= 120 + 40 + 13 + 4 + 1
= 178
118 < 178
Hence index of the greatest power (IGP) of 24 contained in 360! = 118
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