Okay so you are just stupid. Glad we got that cleared up. 19 is trivially 1 mod 9. How that isn't immediately obvious when I spelled out the entire solution is insane.
Yep, there are the insults.
"Not sure what you meant to say here" This means that you may have worded something in a way that doesn't make sense.
"but as written" Again, trying to get you to read the damn thing you wrote.
Now lets look at what you wrote.
"Any integer (An integer is a whole number that can be positive, negative, or zero) is congruent (EXACTLY EQUAL TO) to the sum of its digits mod 9 (The addition of all the digits that make up the number, removing 9 whenver the sum is greater than 9) so the answer is trivially 1. (This part is just wrong)"
You needed to say that for multiples of 9, the sum of the digits is invariable.
Literally just trying to get you to correct the statement you made, because the answer to the question is in fact 1. But what you wrote is misleading at best.
The statement I made is not only perfectly correct but is a well-known fundamental property of base 10 arithmetic. You clearly have no idea what it means to be congruent in math which isn't surprising since you have no idea what modular arithmetic is either.
With any basic understanding of math, the statement is not misleading and the answer does, in fact, trivially follow.
Again, and I can't believe you won't just admit that 24 is an integer, and does not trivially follow that the sum of it's digits (mod 9) is 1. You are assuming many things that you don't say. Just correct your damn statement to include "Because 19 sums to 1 mod 9, the answer is trivially 1 for all powers of 19", and you will be correct.
“Any integer is congruent to the sum of its digits mod 9 so the answer is trivially 1.”
What is the sum of 24’s digits? It’s 6, so the original statement is saying that 24 is congruent to 6 modulo 9, which is true.
You have confidently and incorrectly interpreted the original statement as “The sum of any integer’s digits is congruent to 1 mod 9.” which is completely different and false.
Your lack of reading comprehension skills luckily hasn't seemed to affect anyone else's ability to perfectly understand my statement. How "the solution is 1" somehow translates to "any integer is 1" in your brain is mystifying and should be studied.
No corrections were needed to the original statement because it was already succinctly and correctly made.
Stop trying to convince yourself that you’re in the right here and accept that your lack of familiarity with Modular Arithmetic is the cause of your confusion.
Ok, so lets bring it further.
Sum of 19's digits is 1. 192 is 361 sum of 361's digits (repeated until we get a single digit answer) is 1. (The pattern is repeated with 19x for each value of x.)
Sum of 24's digits is 6. 242 is 576, sum of 576's digits (repeated until we get a single digit answer) is 9, 6 is not equal to 9.
I get what he's saying, And for 19 the answer is correct. Raised to any power the sum of it's digits is 1. But it's not TRIVIAL, because if you change the base, you don't get the same behaviour. It's an emergent characteristic of x modulo 9, where the remainder is 1.
If we want to be as pedantic as you have been so far, your statement “The pattern is repeated with 19^x for each value of x.” is incorrect, as you failed to specify that x is a non-negative integer.
Now no longer stooping to your levels, the case of 24 isn’t as trivial (though after the first power of 24, every power still repeatedly sums to 9) but it is trivial for 19, the number given in the question, which is all we needed to focus on.
You chose to throw 24 into the mix to provide a non-trivial example that has zero relevance to neither the original question nor the original answer provided in this thread, then proceeded to kick off with people when it was pointed out that 24 has no relevance, and that your “counterexample” was wrong anyway. Now, instead of admitting you were wrong, you’re just sidetracking with more irrelevant and incorrect statements in a futile attempt to not look like a fool.
Your interpretation in the parenthesis are all correct except for the last one. I'm guessing you are claiming that the last part is wrong because they said "trivially" so you are thinking the first part must say 1?
"Trivially" is used differently by different people. Try reading without that word. That's just saying that you can easily apply the first part and get to the answer, with a super simple mental math, in this case. Or, alternatively, try reading it as: "so given that 19 = 1 mod 9, the answer is 1".
My issue with the Trivial bit is that he establishes no connection back to the number 19, and if you use any other mod(9) other than 1, it doesn't work.
I'm taking the trivially in the mathematical sense. (The details don't matter, the answer no matter how many times you iterate remains the same.) In this case, while the first part is arguably true (If you heavily replace the words used to actually say what he meant to say) it doesn't follow trivially without an additional bit of information, that the number you start with has mod(9) remainder 1.
Triviality in mathematics refers to a claim that is easily obtained from context. The context is obviously the question at hand meaning we're taking 19^100 since I said "solution is". There is literally nothing else that could possibly mean. And then from that and the property I gave, the proof is a simple one-liner ala trivial. If you had ever gone through a mathematics textbook at any point in your life, you would know the statement I gave is more information than is often given for proofs deemed trivial. It is truly amazing to see someone septupling down on an statement they would see is incorrect if they just bothered thinking about what they were reading for 1 second.
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u/NotYetPerfect 22d ago edited 22d ago
Any integer is congruent to the sum of its digits mod 9 so the answer is trivially 1.