JEEOlympiad
Determine the largest integer k such that 5^k divides the binomial…
Question
Determine the largest integer k such that 5^k divides the binomial coefficient C(2025, 405).
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Step-by-step solution
By Kummer's theorem, the exponent of the prime 5 in C(2025, 405) equals the number of carries when adding 405 and 1620 (= 2025 − 405) in base 5.
Convert: 405 = (3110)_5 since 405 = 3·125 + 1·25 + 1·5 + 0.
And 1620 = (22440)_5 since 1620 = 2·625 + 2·125 + 4·25 + 4·5 + 0.
Aligning (405 = 03110, 1620 = 22440) and adding base 5 from the right: units 0+0=0 (no carry); next 1+4=5 → write 0 carry 1; next 1+4+1=6 → write 1 carry 1; next 3+2+1=6 → write 1 carry 1; next 0+2+1=3 (no carry).
Number of carries = 3.
Hence v_5 = 3.
Final answer3
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