Consider the number 378, 000. (a) Count the number of positive integral factors. (b) Of these factors, how many are even? (c) How many are divisible by both 3 and 7? (d) How many zeros are at the end of 378, 000!

Step-by-step explanation:

A)   2^3 * 3^3 * 5 ^ 2 * 7 ^ 1  

      Total factor of 378,000 including  itself and 1  is

         (3+1)*(3+1)*(2+1)*(1+1) = 4*4*3*2 = 96

B) Knowing its prime factorization we can solve this problem

   378,000 = 378* 100

                  = 2 *27*7*10*10

                  = 2*3*3*3*7*5*2*5*2

                  All the above are indivisible  by any number except itself and one, or are now prime numbers,from here we regroup

                 = 2*2*2*3*3*3*5*5*7

  This can be given as 2^3 * 3^3 * 5 ^ 2 * 7 ^ 1

                making the x= 0 for 2 ^ x

                it becomes 2^0 * 3 ^ 3 * 5 ^ 2 * 7 ^ 1

                 We want the numbers of factors of the above

Therefore (0 + 1 )( 3+1)(2+1)(1+1) = 1 *4*3*2 = 24

C)  From answer (B)   2^0 * 3 ^ 3 * 5 ^ 2 * 7 ^ 1 Is a factor of 378,000

   And from it  three the power of three and 7 are been added to know the exact time 3 and 7 can successfully divide 378,000

3+1 = 4

D) Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow

378,000 Have three trailing zeros

Number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula

+

Solution:

378,000!

+ =  75,600 + 15,120 = 90,720 trailing zeros

the solution is x=1

step-by-step explanation:

we are given our   equation as

let's assume

left side function is

right side function is

now, we can draw graph

and then we can get intersection point

the intersection point will be solution

so, the solution is

x=1