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The number of zeros is determined by how many times 12 divides into 15!. To determine this, since 12 = 22*3, we will look at how many times 2 and 3 divide into 15!.

2 divides 2,6,10,14 once, 4,12 twice, and 8 three times, so that 2 divides 15! exactly 11 times. 3 divides 3,6,12,15 once and 9 twice, so that 3 divides 15! exactly 6 times.

From this, we can see that 12 divides into 15! exactly 5 times, since 125=210*35. In other words, the base-12 representation of 15! ends in 5 zeros.

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Q: How many zeroes are there in 15 factorial in base 12?

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