what is the sum of all natural number values of x such that 15*x = 45 ?

[mathdaughter]

I need your help on the following quiz:

Let a*b = the least common multiple of a and b. What is the sum of all natural number values of x such that 15*x = 45?


45 = 3*3*5
15 = 3*5
then x must have a factor of 3*3
so ???

 

[mmm4444bot]

I need your help on the following quiz

Do you have permission to seek outside help on this quiz?

Let a*b = the least common multiple of a and b. What is the sum of all natural number values of x such that 15*x = 45?

Please check that you have typed the given information correctly. These two statements do not seem to go with one another.

There is only one value for x that makes sense.

 

[mathdaughter]

Do you have permission to seek outside help on this quiz?




Please check that you have typed the given information correctly. These two statements do not seem to go with one another.

There is only one value for x that makes sense.

Yes, I have permission to seek help outside, I think. It is from a book's Try it yourself section. It says the question is originally from 2002 Mathcounts Handbook.

Yes, I have checked again. I have typed in all the information. It is just a copy of the words on my book. The answer is 54 but I don't know how we can get the 54.

 

[stapel]

Let a*b = the least common multiple of a and b. What is the sum of all natural number values of x such that 15*x = 45?

Huh. Weird.

The "Let" statement and the "What" question don't appear to have any connection, but there's at least one place online (here, page 10, question 115) where one can see that, yes, indeed, they really are supposed to go together... somehow. (And, yes, the answer key, on page 16 of the same document, confirms that the answer is supposed to be "54".) :shock:

The actual Handbook (for 2001-2002) has the exercise on page 20, but the solutions page (on page 21) just lists the answer; it does not explain how one is expected to use the given information to arrive at that answer. However...

Many times in these Handbooks (I've noticed, while trying to find the above information), exercises use "*" to mean "an operator that we will now define", rather than "times". So what I think is going on is the following:

---

7. We will define the operation "@" as follows:

. . . . .a @ b = c

...where c is the Least Common Multiple of a and b. Under this definition, find the sum of all natural numbers x such that:

. . . . .15 @ x = 45

That is, find the sum of all natural numbers x such that the LCM of 15 and x is 45.

---

If this is a correct understanding (and, if so, shame on the authors for using such confusing notation!), then we have:

. . . . .45 = 3 * 3 * 5

. . . . .15 = 3 * 5

The only way 15 and x will have a LCM of 45 is if the following is true:

LCM:

 15 : 3     * 5
  x : ? * ? * ?  
----+---------------
LCM : 3 * 3 * 5 = 45

(For an explanation of the method being used above, go here.)

This means that x must have two copies of the factor 3 (so as to get the second copy for the LCM of 45), and maybe a copy of 5 (or not). So the only possible x-values would be 9 (which has that second copy of 3, but no copy of 5) and 45 (which has the second copy of 3, and also a copy of 5). These values sum to 54.

I'm not an advocate of giving out the answers, but this question was formatted so confusingly that I felt the need to make an exception. Comments and corrections are welcome!

 

[mathdaughter]

Thank you for your explanation. Learned a lot from you.

 

[JeffM]

To stapel,

I studied abstract algebra back when mastodons roamed what now is Manhattan (they were a menace to students at Columbia). At that time, the asterisk was frequently used to represent any binary operation. I fully agree with you that the now ubiquitous use of the asterisk to represent multiplication means that it should no longer be used to denote any other binary operation.

 

[ksdhart2]

To stapel,

I studied abstract algebra back when mastodons roamed what now is Manhattan (they were a menace to students at Columbia). At that time, the asterisk was frequently used to represent any binary operation. I fully agree with you that the now ubiquitous use of the asterisk to represent multiplication means that it should no longer be used to denote any other binary operation.

I'm in an abstract algebra class right now and we're learning about rings. The professor prefers to use an x with a circle around it to denote "special" multiplication within the ring.