To solve the equation

\[ 3^x + 3^y + 3^z = 837, \]

we begin by factoring 837. First, we divide 837 by 3:

\[ 837 \div 3 = 279, \quad 279 \div 3 = 93, \quad 93 \div 3 = 31, \]

which gives the factorization

\[ 837 = 3^3 \times 31. \]

We now look for powers of 3 that sum up to 837. Using trial and error with integer exponents, let’s start with these values:

\[ 3^6 = 729, \quad 3^5 = 243, \quad 3^4 = 81. \]

Now, testing these values, we find:

\[ 3^6 + 3^5 + 3^4 = 729 + 243 + 81 = 837. \]

Thus, the solution is:

\[ x = 6, \quad y = 5, \quad z = 4. \]

## 8 thoughts on “Find exponents for $\Huge 3^x + 3^y + 3^z = 837$”

Find integers for \[ x,y,z \] that satisfy the equation or prove that it cannot be done. \[3^x+3^y+3^z=837\]

Note* We should know what course we are in prior to attempting this.

One could use trial and error since 3 + 3 + 3 = 9, and 3^1 + 3^2 + 3^3 = 39 and then reverse engineer a solution. But there are other solutions from various courses already in play that could be used.

The solution above is incorrect. It doesn’t use the factorization it finds and just says to guess and check. It has not proved that this is the only solution (indeed clearly so, as any permutation of the values among the variables is a valid solution and the fact that the given solution doesn’t work).

Instead we should divide the equation by $3^3$ giving $$3^{x-3}+3^{y-3}+3^{z-3} = 31$$

It is clear that no exponent can be negative so we need only consider non-negative solutions. Then as all powers of 3 with the exponent greater than or equal to 1 are multiples of 3 (and any sum of multiples of 3 is a multiple of 3), we must have that one of the exponents is zero. Without loss of generality take this to be $x$. Then we have $x=3$. This then reduces the equation to $$3^{y-3}+3^{z-3} = 30$$. Then as $3^4$ > 30, we must have that both exponents are less than 4. Enumerating all possible combinations gives that without loss of generality $y-3=3$ and $z-3=1$. However as the situation is entirely symmetrical we can permute the exponents however we wish amongst the variables. Thus, the solution set is given by $$(x,y,z) \in {(3, 6, 4), (3,4,6),(4,3,6),(4,6,4),(6,3,4),(6,4,3)}$$

Excellent point. However, the question posed did not specifically ask for the use of a particular solution method. You’re assuming it required a demonstration of all possible solutions using a single technique, but it simply asked for the values that satisfy the equation. In different contexts, such as a number theory course or a related subject, the approach could range from applying a full theorem to making a well-reasoned guess.

In this instance, I adapted the explanation for ChatGPT based on the student’s request since we’re testing the layout. However, for your solution to be considered fully correct, you must reference the theorem or rules utilized. Otherwise, it could be interpreted as just a lucky guess. A valid solution requires first acknowledging the definitions, theorems, and rules before applying them. Interestingly, the arithmetic itself is the least critical part of a question nowadays, as computers can easily handle that. The conceptual understanding is what truly matters.

Happy to see some interaction on here! Thanks for showing up!

Jonathan

Also, don’t forget your braces to say an object is in a set. To parse them, put a backslash in front and you can use left/right to make them match the size. Each thing you write is an English sentence and requires proper grammar and punctuation.

I did actually include curly braces (including back slashes), but for some reason even though it is rendering correctly in my own LaTeX installation, here it is not. The website also isn’t displaying inline equations, but maybe there is just slightly different syntax for this site.

Ah. Yes, the wordpress and plug for latex may be a bit off. I will play with it. I know the braces usually need [ { } ]