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25 Jan 2026, 22:00
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C
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Difficulty:
45%
(medium)
Question Stats:
62% (01:27) correct
38%
(01:25)
wrong
based on 198
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What is the highest integer k for which 125^k perfectly divides 200! ?
A. 49
B. 48
C. 16
D. 13
E. 1
A. 49
B. 48
C. 16
D. 13
E. 1
25 Jan 2026, 23:24
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Let's use Legendre's formula to calculate the highest power of a prime 'p' that divides n! for this question. Since 125^k = 5^(3k), we will calculate the highest power of 5 in 200!
200/5 + 200/25 + 200/125 = 40+8+1 = 49
Thus, the highest power of 5 in 200! is 49. However, the question wants us to find the highest power of 125 in 200!:
5^49 = 5^3k
Thus, 3k = 49; the highest integer k for which this would be possible = 16
Hence, the correct answer is Option C - 16
Hope this helps!
200/5 + 200/25 + 200/125 = 40+8+1 = 49
Thus, the highest power of 5 in 200! is 49. However, the question wants us to find the highest power of 125 in 200!:
5^49 = 5^3k
Thus, 3k = 49; the highest integer k for which this would be possible = 16
Hence, the correct answer is Option C - 16
Hope this helps!
