A sum of the digits of the exponential variables of (2^x)(3^y)(5^z) must equal a desired value and the expression product between some range -- how?

[Bill_Graham]

I have a math problem I don't know how to approach. Seems similar to one that I could find answered on the internet, where you have (2^x)(3^y)(5^z), and the question was how many combinations of factors will this possibly produce, the answer was (x + 1)(y + 1)(z + 1), and which makes sense. NOW I have been given a more difficult challenge. Here is the problem I must solve, and I don't know how to do it clean and neat and without the help of a calculator. (I am not allowed to use a calculator.) So here goes:
Given that we have an exponential expression (2^x)(3^y)(5^z).
Limiting the product of the exponential expression to a number between 5,000 and 6,000.
And, we will look for a sum of digits of the exponents, x, y, and z, such that x + y + z = either "5" or "8" or "11" or "18"... any one of these.

I won't bore you with my convoluted thinking on this matter. And, yes, I know, I can probably take a couple of days of time to grind though this and learn from the experience, but I still would not learn the generalized solution that I am sure someone has already learned and can share with me here. I have never run into a problem quite like this. I don't see a practical application, but it is probably a good thing to know, I suppose. So if anyone can answer this question with a nice neat generalized methodology, not involving the use of calculators (just maybe some scratch paper and some simple/clever math that way), maybe also they could point out to me the skill I am supposed to learn from knowing how to solve such a problem.

Thanks, and I can't wait to see what it is I have not thought of. (Maybe the "skill" I need is to be able to take square roots of such numbers as 5,000 or 6,000 or anything in between right out of my head? Just a thought.)

 

[Deleted member 4993]

I have a math problem I don't know how to approach. Seems similar to one that I could find answered on the internet, where you have (2^x)(3^y)(5^z), and the question was how many combinations of factors will this possibly produce, the answer was (x + 1)(y + 1)(z + 1), and which makes sense. NOW I have been given a more difficult challenge. Here is the problem I must solve, and I don't know how to do it clean and neat and without the help of a calculator. (I am not allowed to use a calculator.) So here goes:
Given that we have an exponential expression (2^x)(3^y)(5^z).
Limiting the product of the exponential expression to a number between 5,000 and 6,000.
And, we will look for a sum of digits of the exponents, x, y, and z, such that x + y + z = either "5" or "8" or "11" or "18"... any one of these.

I won't bore you with my convoluted thinking on this matter. And, yes, I know, I can probably take a couple of days of time to grind though this and learn from the experience, but I still would not learn the generalized solution that I am sure someone has already learned and can share with me here. I have never run into a problem quite like this. I don't see a practical application, but it is probably a good thing to know, I suppose. So if anyone can answer this question with a nice neat generalized methodology, not involving the use of calculators (just maybe some scratch paper and some simple/clever math that way), maybe also they could point out to me the skill I am supposed to learn from knowing how to solve such a problem.

Thanks, and I can't wait to see what it is I have not thought of. (Maybe the "skill" I need is to be able to take square roots of such numbers as 5,000 or 6,000 or anything in between right out of my head? Just a thought.)

Are there constrains on x, y and z (like x, y & z are integers or x, y & z >0, etc.)?

 

[Dr.Peterson]

I have a math problem I don't know how to approach. Seems similar to one that I could find answered on the internet, where you have (2^x)(3^y)(5^z), and the question was how many combinations of factors will this possibly produce, the answer was (x + 1)(y + 1)(z + 1), and which makes sense. NOW I have been given a more difficult challenge. Here is the problem I must solve, and I don't know how to do it clean and neat and without the help of a calculator. (I am not allowed to use a calculator.) So here goes:
Given that we have an exponential expression (2^x)(3^y)(5^z).
Limiting the product of the exponential expression to a number between 5,000 and 6,000.
And, we will look for a sum of digits of the exponents, x, y, and z, such that x + y + z = either "5" or "8" or "11" or "18"... any one of these.

I won't bore you with my convoluted thinking on this matter. And, yes, I know, I can probably take a couple of days of time to grind though this and learn from the experience, but I still would not learn the generalized solution that I am sure someone has already learned and can share with me here. I have never run into a problem quite like this. I don't see a practical application, but it is probably a good thing to know, I suppose. So if anyone can answer this question with a nice neat generalized methodology, not involving the use of calculators (just maybe some scratch paper and some simple/clever math that way), maybe also they could point out to me the skill I am supposed to learn from knowing how to solve such a problem.

Thanks, and I can't wait to see what it is I have not thought of. (Maybe the "skill" I need is to be able to take square roots of such numbers as 5,000 or 6,000 or anything in between right out of my head? Just a thought.)

I'd like to know the source of the problem. It sounds more like a contest problem (intended to challenge your ability to solve non-routine problems) than a homework problem (intended to teach you something). It doesn't look practical, or like a special case of a useful general idea.

Also, I don't think you've quoted the problem as given to you, which might be a little different from what I am thinking it is. (I assume that x, y, and z have to be non-negative integers, but they might instead have to be positive, for example; and it isn't clear whether you just want to find one such sum, or all.)

I see no clever way to go straight to an answer; all I'd do to start with would be to "play" with the problem to get a feel for how things work. The combination of a restriction on the range of the value of the expression, and a restriction on the sum of the exponents seems odd (especially the choice of permitted sums), eliminating nice things I might otherwise want to do.

So (and I'm just thinking as I write, with no solution) I might start with the sum 5 and see whether I can make any number at all in the right interval. That gets me thinking about several things that might be useful later. But to do any more, I really need to know exactly what the problem really is.

 

[Deleted member 4993]

If x, y & z are integers or x, y & z >0 , the possible numbers (with prime factors of 2,3 & 5) must be divisible by 10. Moreover, those must be divisible 3. That reduces the numbers to investigate further to ≤34. This is do-able by hand (if all the ifs become yes).