the product of 1000 whole number is 1000, what is the largest possible value the sum of these numbers can have
what does "product of 1000 whole number" mean?
Multiply 1 times 1 999 times and then multiply that product by 1,000. Your product is 1,000 and the sum is 1,999.
Well, let's think about this. Since the product of the numbers is 1000, we know that we have a factor of 1000 in there somewhere. So, we need to distribute the sum as evenly as possible to get the largest possible value.
If we divide 1000 by 2, we get 500. So, one of the numbers could be 500. Then, if we divide the remaining 500 by 2 again, we get 250. So, another number could be 250.
If we continue this process, we can see that the largest possible value the sum of these numbers can have is when all the numbers are as close to each other as possible. In this case, the sum would be 500 + 250 + 125 + 62 + 31 + 16 + 8 + 4 + 2 + 1 = 999.
So, the largest possible value the sum of these numbers can have is 999, just one short of a perfect 1000. Almost there!
To find the largest possible value for the sum of the whole numbers, we should distribute the product as evenly as possible across the numbers.
Let's assume we have 'n' number of whole numbers. We need to find the 'n' numbers whose product is 1000.
The prime factorization of 1000 is 2^3 * 5^3. To distribute the factors as evenly as possible, we can assign the highest factors to the smallest numbers.
So, we should assign 2^3 to the first number, 2^3 to the second number, and so on until the nth number. After reaching the n-1th number, we should assign the factors 5^3.
If we assign 2^3 to all the numbers, we can find the value of n by calculating 1000^(1/3). Rounding this number to the nearest whole number gives us n.
Let's calculate this:
n = 1000^(1/3)
n = 10
So, we have 10 numbers whose product is 1000. Now we can distribute the factors as described earlier.
The first 3 numbers would have 2^3 allocated to them, and then all subsequent numbers would have 5^3 allocated to them.
Therefore, the largest possible value for the sum of these numbers is:
2^3 + 2^3 + 2^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 = 8 + 8 + 8 + 125 + 125 + 125 + 125 + 125 + 125 + 125 = 837.
So, the largest possible value for the sum of these numbers is 837.
To find the largest possible sum, we need to maximize the individual numbers while still keeping their product as 1000.
First, let's prime factorize 1000:
1000 = 2^3 * 5^3
To maximize the sum, we need to distribute these prime factors evenly among the whole numbers. We want to have as many large factors as possible. Let's start with the largest prime factor, 5:
5 x 5 = 25
5 x 5 = 25
5 x 5 = 25
Now, let's move on to the next largest prime factor, 2:
2 x 2 x 2 = 8
2 x 2 x 2 = 8
2 x 2 x 2 = 8
To maintain a product of 1000, we need to keep the same number of prime factors for every one of the 1000 numbers.
So, for the sum of the numbers to be maximum, we can have:
25 + 25 + 25 + 8 + 8 + 8 + ... (repeating 1000 times)
Since we have 1000 numbers, we can divide the numbers into groups of 8 and have 125 such groups.
Each group sums up to 3 numbers of 25 + 8 = 33.
Therefore, the largest possible sum of these 1000 numbers is:
125 x 33 = 4125.