SOLUTION: What is the greatest power of 7 that divides into 1234!, with no remainder?

Question 1104111: What is the greatest power of 7 that divides into 1234!, with no remainder?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The question is equivalent to asking how many factors of 7 there are in 1234!.

1 of every 7 numbers from 1 to 1234 has at least one factor of 7: 1234/7 = 176.
1 of every 7 of those 176 numbers has at least one more factor of 7: 176/7 = 25.
1 of every 7 of those 25 numbers has yet one more factor of 7: 25/7 = 3.

The total number of factors of 7 in 1234! is 176+25+3 = 204.

So 7^204 will divide evenly into 1234!, but 7^205 will not.

Note the process here is applicable to any similar problem where the divisor is a prime number.

So, for example, if you wanted to now the largest power of 13 that would divide evenly into 1234!, the answer would be...
1234/13 = 94
94/13 = 7
94+7 = 101

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