SOLUTION: there are six different numbers between 1 and 10 in the form of the form 3m+2n. what is the sum of these six numbers?

Question 1090370: there are six different numbers between 1 and 10 in the form of the form 3m+2n. what is the sum of these six numbers?
Answer by greenestamps(13367) (Show Source): You can put this solution on YOUR website!
The problem is not well defined; I can't see any interpretation that gives 6 different numbers between 1 and 10 that can be written in the prescribed form.
(1) I assume by "numbers" you mean whole numbers.
(2) Does "between 1 and 10" include both 1 and 10? or neither?
(3) What are the restrictions on m and n?
If m and n must be positive integers, then the answer is less than 6.
If m and n can be either positive integers or 0, then every number between 1 and 10 except 1 can be written in the prescribed form.
If m and n can also be negative integers, then every integer can be written in the prescribed form.

I am new to this web site; so I don't know the right way to respond to your message.... So I'm adding it to my original response.

Nobody can help you find the answer until you provide a precise description of the problem. You have said that 1 and 10 do not count; that clears up one ambiguity. Presumably we are looking at whole numbers, so we are considering 2, 3, 4, 5, 6, 7, 8, and 9.

Your statement of the problem says we are to find the sum of the 6 numbers in this group that can be expressed as 3m+2n -- but you say nothing about what values m and n can have.

Again I assume m and n can have only integer values; otherwise clearly any number can be expressed in the form 3m+2n.

And if m and n can be negative integers, than any integer can be expressed in the form 3m+2n.

So there seem to be only two possibilities: (1) m and n have to be positive integers, or 0; or (2) m and n have to be positive integers.

But neither of these interpretations gives 6 numbers between 1 and 10 that can be written in the form 3m+2n.

If m and/or n can be 0, then we have

2 = 3(0)+2(1)
3 = 3(1)+2(0)
4 = 3(0)+2(2)
5 = 3(1)+2(1)
6 = 3(2)+2(0) or 3(0)+2(3)
7 = 3(1)+2(2)
8 = 3(2)+2(1) or 3(0)+2(4)
9 = 3(3)+2(0) or 3(1)+2(3)

So all 8 of the numbers between 1 and 10 can be written in the required form.

If neither m nor n can be 0, then we have

5 = 3(1)+2(1)
7 = 3(1)+2(2)
8 = 3(2)+2(1)
9 = 3(1)+2(3)

And in this case only 4 numbers can be written in the required form.

So again I say, no interpretation of the problem I can see will give us 6 of the 8 numbers being able to be expressed as 3m+2n.

If you want help with this, you need to make sure you are stating the problem correctly, including saying what values m and n can have.
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