The product of 2 positive integers is 1000. What is the smallest possible sum of these 2 integers?
possibilies
1x1000
2x500
4x250
5x200
8x125
10 x 100
20x50
25x40
40x25 --- repeating from here on
so the smallest possible sum of those factors
= 25 + 40 = 65
OR
by the "Just Think About It" theorem, the two number would have to be as close as possible to each other, that is, near the √1000
which is appr 31.6
the closest number to 31.6 which divides evenly into 1000 is 25, so the other must be 40
Well, I must say, finding the smallest possible sum of two positive integers whose product is 1000 is like trying to find the smallest dinosaur or the shortest giraffe. It's quite a tricky task, isn't it?
But fear not, my friend! I am here to assist you in this amusing mathematical conundrum. Let's put our thinking caps on and embark on this delightful journey together.
Now, to find the smallest possible sum, we need to look for two positive integers that have a product of 1000. Let's start with the smallest possible positive integer, which is 1, and see if we can find a pair that meets our criteria.
Hmm... let's try 1 and 1000. While their product is indeed 1000, the sum would be 1001. Oh my, that doesn't quite fit our requirement, does it?
What about 2 and 500? Again, their product is 1000, but the sum would be 502. Close, but no banana!
Let's continue our little game. How about 4 and 250? Hey, we're getting there! Their product is still 1000, and the sum would be 254. Progress!
Let's keep going. How about 5 and 200? Nope, not quite right. The product is 1000, but the sum is 205.
Ah, there we have it! The magic pair is 10 and 100. Their product is indeed 1000, and the sum is a charming 110! Ta-da!
So, my friend, the smallest possible sum of two positive integers whose product is 1000 is 110. Voilà!
To find the smallest possible sum of two positive integers whose product is 1000, we need to find the pair of positive integers whose sum is minimized. One way to do this is to find the pair of positive integers whose difference is minimized.
Let's start by finding the factors of 1000:
1 × 1000 = 1000
2 × 500 = 1000
4 × 250 = 1000
5 × 200 = 1000
8 × 125 = 1000
10 × 100 = 1000
20 × 50 = 1000
From these pairs, we can see that the pair with the smallest difference is 20 × 50 = 1000. Therefore, the smallest possible sum of these two integers is 20 + 50 = 70.
To find the smallest possible sum of two positive integers whose product is 1000, we need to find the pair of positive integers that multiply to 1000 and have the smallest sum.
One way to approach this problem is to factorize 1000 into its prime factors: 2 * 2 * 2 * 5 * 5 * 5. From this factorization, we can see that the product of two positive integers is 1000 when both integers contain the factors of 2 and 5.
To find the smallest possible sum, we should pair the factors as closely as possible. In this case, we can pair two 2s and one 5 to get the smallest sum:
2 * 2 * 5 = 20
Therefore, the smallest possible sum of two positive integers whose product is 1000 is 20.