Find the number of positive integers n such that the highest power of 7 dividing n! is 8.
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Given : positive integers n such that the highest power of 7 dividing n! is 8.
To find : Value of n
Solution:
highest power of 7 dividing n! is 8.
=> [n/7] + [n/7²] + [n/7³] + .+ .+ .+ . + = 8
7² = 49 if n < 49 then
We get only [ n/7] where n < 49
Hence [ n/7] < 7
We need highest power = 8
Lets check for 49
= [49/7] + [49/7²] + [49/7³] + .+ .+ .+ . +
= 7 + 1 + 0 + 0 +
= 8
Hence n = 49 Satisfy this
highest power of 7 dividing 49! is 8
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