Number System: Factorials & No. of Zeros Test-1

Since 7is prime number so we will find only number of sevens in 39!
So when we divide the 39 with 7 we get 5 as quotient and 4 as the remainder ,
since now quotient is smaller than the divisor and cannot be be divided further so the total number of sevens in 39!  are 5
That means the highest power of 7 that divides the 39! Is 7⁵ ,
so (d) is the right answer

Since 8 can be written as 2 x 2 x 2 =2³
So we will make  pairs of 2³ from the number of 2’s in 88!
Now the number of two’s in 88!  are 85
And the number of pairs of 2³ are 85/3= 28 with one 2 remainder
so the maximum pair that can be made are 28 and finally the maximum power of 8
that can divide the 88! So the right choice for this question is (a)

24 can be written as  2 x 2 x 2 x 3 = 2³ x 3
So we will make pairs of 2³ x 3 from the number of two’s and number of threes ‘s in 80!
The number of 2’s in 80!  are 78 ;
Number of 8’s in 80 ! is 78/3 = 26
The number of 3’s in 80 ! are 36
So the limiting power in 24  = power of 8 = 26
the highest power of 24 that can divide the 80! is 26.

To solve this question we have to remember some points that the number of zeros are depends upon the number of pairs of 2 x 5 We know that number of zero in 5! = 1
5! to 9! = 1
10! to 14 ! = 2
15! to 19 ! = 3
20! to 24 ! = 4
25! to 29 ! = 6
Here the order of zero shifted to 4 to 6 because in 25!  ,
25 it self is square of 5 i.e. 25 = 5 x5 therefore there are  six 5’s in 25!
so the number of zeros are 6 . Come to the question
So we know that the number of zeros in 100!  are 24
101! to 104 ! = 24
105! to 109 ! = 25
110! to 114 ! = 26
115! to 119 ! = 27
120! to 124 ! = 28
125! to 129 ! = 31
From this 125 is a multiple of 25 and can be written as 5 x 5x 5
So there will be the addition of 3 more zeros in it so numbers of zeros will shifted to 28 to 31.
Now see the options , option number (c) is not possible in any case

Initially 45 can be written as = 3 x 3 x 5 ,So we need pair of 3² x 5
For this lets see the number of 3’s and 5’s in 45!
The number of 3’s in 45! are 21
Number of 5’s in 45 ! are 10
So we can make only 10 maximum pairs of 3² x 5
So the number of zeros in 45! are 10
Now we are asked for the number of zeros in (45!)^4! So if in one 45! There are 10 zero
Then in 24 , 45! There would be 240 zeros
So the best answer for this question is option (d).