Question

Find the number is positive integers n such that the highest power of 7 dividing n! is 8.​

Answer 1

Topic :-

Binomial Theorem

Given :-

The highest power of 7 dividing n! is 8.

To Find :-

The number of positive integers 'n' which satisfies given statement.

Solution :-

So, we need to find 'n' for which the highest power of 7 dividing n! is 8.

Calculating highest power of a number in a factorial.

If 'p' is prime number, then the highest power of 'p' in a factorial 'n' is given by :-

$\left[\dfrac{n}{p} \right ] + \left[\dfrac{n}{p^2} \right ] + \left[\dfrac{n}{p^3} \right ] + . . . . . . . . . . . .$

Putting values  of 'p' as 7.

$\left[\dfrac{n}{7} \right ] + \left[\dfrac{n}{7^2} \right ] + \left[\dfrac{n}{7^3} \right ] + . . . . . . . =8$

Using Hit and Trial Method for value of n :-

Put 'n' = 48,

6 + 0 + 0 + . . . = 6

So, equation do not get satisfied.

Put 'n' = 49, 50, 51, 52, 53, 54 and 55.

7 + 1 + 0 + 0 + . . . . = 8

So, equation get satisfied.

Put 'n' = 56,

8 + 1 + . . . . . = 9

So, equation do not get satisfied.

So,

Numbers from 49 to 55 will have highest power of 7 as 8 while dividing n!.

Answer :-

So, total 7 positive integers are there which satisfy the statement.

Note : It will not include integers from 1 to 48 because they do not have 8 as highest power of 7 dividing them.

Answer 2

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 also

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 highest 8 (8 or less than 8) hence 55 integers satisfy

from 49! to 55!  power of 7 is exactly 8 ( hence 7 integers satisfy)

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