# maths

posted by .

the sum of N positive integers is 19. what is the maximum possible product of these N numbers?

thx.

Nice problem.
Let's look at some cases

1. N=2
clearly our only logical choices are 9,10 for a product of 90
It should be obvious that the numbers should be "centrally" positioned
e.g 2 and 17 add up to 19, but only have a product of 34

2. N=3
I considered 4,7,8 for a product of 224, 5,6,8 for a product of 240 and 2,8,9 for a product of only 144.

3. N=4

You should quickly realize that we can't start too high or we will go over 19
so 2,3,4,10 ---> product 240
2,3,5,9 ---> product 270
2,3,6,8 ---> product 288
3,4,5,7 ---> product 420 *!*!

4. N=5

We have to start with 1,2,3 or else we run over,
e.g. if we start with 2,3,4,5 we already have a sum of 14, so we need a 5, but we already used it.

so the only choices would be
1,2,3,4,9 ---> product 216
1,2,3,5,8 ---> product 240
1,2,3,6,7 ---> product 252

so it looks like the 4 numbers 3,4,5,7 which have a sum of 19 and a maximum product of 420 are it!

okay i get it!so you will just have to try out all the possible way!thanks for the help~

Do the numbers have to be different?

• maths -

If the numbers can be repeated:
3*(2^8) = 768
3+(2*8) = 19

## Similar Questions

1. ### discrete math

use a direct proof to show that the product of two odd numbers is odd. Proofs: (all the nos. i used are odd) 3 x 3 = 9 5 x 9 = 45 7 x 3 = 21 Yes, but you didn't prove the statement for "all" odd integers, only the odd integers you …
2. ### calculus

find two positive integers such that the sum of the first number and 4 times the second number is 1000 and the product of the numbers is as large as possible
3. ### math

can you answer this question in a different and more logical way than this method below: we will examine the sum of cubes of two numbers, A and B. Without losing generality, we will further assume that A=2nX and B=2n+kY where X is …
4. ### Integers

Suppose a1, a2, . . . , an is a list of n numbers with the following properties: The sum of those n numbers is 500. The sum of the smallest three of those numbers is 48. The sum of the largest two of those numbers is 35. (Note: There …
5. ### ALGEBRA.

The product of 2 positive integers is 1000. What is the smallest possible sum of these 2 integers?
6. ### maths

If the product of two positive integers is 363, and the least common multiple of them is 33, what is the sum of the two positive integers?
7. ### maths

If the product of two positive integers is 363, and the least common multiple of them is 33, what is the sum of the two positive integers?
8. ### MATHS

A national math contest consisted of 11 multiple choice questions, each having 11 possible choices, of which only 1 of the choices is correct. Suppose that 111 students actually wrote the exam, and no two students have more than one …
9. ### Maths

Let x,y,z be non-negative real numbers satisfying the condition x+y+z=1. The maximum possible value of x^3y^3+y^3z^3+z^3x^3 has the form ab where a and b are positive, coprime integers. What is the value of a+b?
10. ### Maths

The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers?

More Similar Questions