Factorial Remainder

Here is how i would do this question

Write the prime factorisation of each option

Take the biggest prime factor of each option and check the highest power that will suffice it for example

In option 1 we can see 11

Now we will

151/11+151/11² we will not take 11³ as it is greater than 131. 151/11 is not a division symbol it is checking how many 11's will suffice so 151/11 is 13 and 151/121 will be 1

13+1=14 I do not know why in the solution it is 13

For option 2 the max prime is again 11 so it will be the same

For option 3 it is 7 so 151/7+151/49 again 7³ is greater than 151 so won't take it

Similarly for option 4

We are checking that what is the highest power of the given option that will divide 151!

Above is the formula used

Shorter method

No need to manually check for each option

Just pick the highest prime from each option

Obviously 151! Will be divisible by a higher power of 7[i.e. contain more seven] as compared to 11.

So indirectly speaking we need to pick the option whose biggest prime factorization is the lowest among all the options

Highest prime factor of each option

A 11

B 11

C 7

D 13

Now the lowest among these is option C which is the answer