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How many trailing zeros does 2026! have when it is written in base…
Question
How many trailing zeros does 2026! have when it is written in base 12? (A trailing zero in base 12 corresponds to a factor of 12 = 2^2 * 3.)
✓ Verified answer: 1009checked by our engine — not a guess
Step-by-step solution
The number of trailing zeros of N!
in base b is the largest k with b^k | N!.
Since 12 = 2^2·3, we need the largest k with 2^(2k)·3^k dividing 2026!, i.e.
k = min( floor(v_2(2026!)/2), v_3(2026!) ).
Legendre: v_2(2026!) = 1013+506+253+126+63+31+15+7+3+1 = 2018, so floor(2018/2)=1009.
v_3(2026!) = 675+225+75+25+8+2 = 1010.
k = min(1009, 1010) = 1009.
Final answer1009
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