Let 3a be the highest power of 3 that divides 1000!. What is a?
498
To find the value of a, we need to determine the power of 3 that divides 1000!. One way to do this is by examining the power of 3 in the prime factorization of each number from 1 to 1000.
Step 1: Count the multiples of 3 from 1 to 1000
There are 1000/3 = 333 multiples of 3 in the range of 1 to 1000.
Step 2: Count the multiples of 9 (3^2) from 1 to 1000
There are 1000/9 = 111 multiples of 9 in the range of 1 to 1000.
Step 3: Count the multiples of 27 (3^3) from 1 to 1000
There are 1000/27 = 37 multiples of 27 in the range of 1 to 1000.
Step 4: Count the multiples of 81 (3^4) from 1 to 1000
There are 1000/81 = 12 multiples of 81 in the range of 1 to 1000.
Step 5: Count the multiples of 243 (3^5) from 1 to 1000
There is only 1 multiple of 243 in the range of 1 to 1000.
Step 6: Count the multiples of 729 (3^6) from 1 to 1000
There is no multiple of 729 in the range of 1 to 1000.
Step 7: Add up the counts from each step
333 + 111 + 37 + 12 + 1 = 494
Therefore, the highest power of 3 that divides 1000! is 3^494. So, a is equal to 494.
To find the highest power of 3 that divides 1000!, we need to determine the number of factors of 3 in the prime factorization of 1000!.
Step 1: Determine the number of multiples of 3 from 1 to 1000.
In this case, we can use integer division to find the highest power of 3 that divides 1000. Dividing 1000 by 3, we get 333 with a remainder of 1. This means there are 333 multiples of 3 from 1 to 1000.
Step 2: Determine the number of multiples of 9 from 1 to 1000.
We need to count the multiples of 9, as each multiple of 9 contributes an additional factor of 3. Dividing 1000 by 9, we get 111 with a remainder of 1. Therefore, there are 111 multiples of 9 from 1 to 1000.
Step 3: Determine the number of multiples of 27 from 1 to 1000.
Similar to step 2, we divide 1000 by 27, and we get 37 with a remainder of 19. Thus, there are 37 multiples of 27 from 1 to 1000.
Step 4: Continue this process until the divisor exceeds 1000.
We will see that we cannot exceed multiple of 27 as there are only 37 multiples till 1000 (For higher power, one more level of factorization is required, but it does not exceed 1000.)
Adding up the counts, we have:
333 + 111 + 37 = 481
Therefore, the highest power of 3 that divides 1000! is 3^481.
Hence, a = 481.