5. Find the number of positive integers n such that the highest power of 7 dividing n! is 8.
Answers
Given : positive integers n such that the highest power of 7 dividing n! is 8.
To find : Number of Such positive integers
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
check for 55
= [55/7] + [55/7²] + [55/7³] + .+ .+ .+ . +
= 7 + 1 + 0 + 0 +
= 8
check for 56
= [56/7] + [56/7²] + [55/7³] + .+ .+ .+ . +
= 8 + 1 + 0 + 0 +
= 9
Hence 56! has power of 7 more than 8
From 1! to 55! power of 7 is max 8
from 49! to 55! power of 7 is exactly 8
Hence 55 positive integers
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Given :-
- positive integers n such that the highest power of 7 dividing n! is 8.
To find :-
- Number of Such positive integers
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
check for 55
= [55/7] + [55/7²] + [55/7³] + .+ .+ .+ . +
= 7 + 1 + 0 + 0 +
= 8
check for 56
= [56/7] + [56/7²] + [55/7³] + .+ .+ .+ . +
= 8 + 1 + 0 + 0 +
= 9
Hence 56! has power of 7 more than 8
From 1! to 55! power of 7 is max 8
from 49! to 55! power of 7 is exactly 8
Hence 55 positive integers