The word "different" was added to the opening post not long ago, as noted by Fudge Muffin in post 17. It was not there when you first said that m being zero excludes k being zero, and when AlphaNumeric corrected you.
The way this was presented it seems to be a home work problem. However, the problem itself is unique, in that it can be answered both from a purely mathematical approach, as has been being discussed or as a matter of logic, more specifically addressing this unique problem. Without knowing what kind of solution is expected, it is hard to know whether the problem is one of math or just logic. If you are expected to provide a purely mathematical proof, the following will be of little help. That said... Threre is a unique similarity in the powers of three of the numbers, 5, 6 & 11. For these three all powers above 0, the units digit is fixed. The number 7 is the odd man out. Knowing that the sum of the units digits of 5^j, 6^k, 7^l & 11^m must equal 6, provides an easy solution for two of the four. Both of the remaining unknowns must be the highest allowable power for the number involved...
What was the answer? And what did you do to work it out. I wrote a macro to recalculate a formulated Excel sheet solve it for me. Just allowed JKLM to be Randbetween (0,4) rather than to worry about trying to make them always be different. There seems to be only one solution.
I went through the combinations with pen and paper Please Register or Log in to view the hidden image! 5^(4,3,2,1,0) 6^(4,3,2,1,0) 7^(3,2,1,0) 11^(3,2,1,0) 4, 0, 2, 3
so to get the answer, you must have awareness of the fact that powers of 5,6 and 11 don't change the units digit... If you had never heard of that fact before, and you had, say, 10 minutes to do the problem, is it possible?
If the problem was one looking for logic and set recognition. The squares should have provided the clue about powers of 5, 6 & 11. You also need to know that ^0 can be one of the powers in the answer. I am not sure this is what your teacher was looking for. It would not work with all similar problems. It is unique to those three numbers 5, 6 & 11.., well 1 would also work but that would be too obvious.., or any numbers ending in 5, 6 or 1, but for powers of double and triple digit numbers, it would not be a simple math problem, to me. When I was in school we were not allowed to use calculators. The squares are no problem, so the first two powers should be easy. That leaves just working out two. I am not sure how fast that would have jumped out for me if Tach had not posted that m < 4. Because from there, powers of 5 are easy, to see. This is more recognizing a trick of numbers than anything else. I would be interested to know two things. First if this is the kind of thing your teacher might have been looking for and second what Tach, Alpha, Guest or Prom would have or did come up with... What passes for simple math today is far different than what was considered simple when I was in school. But recognizing patterns is an important skill in practice. Honestly.., though I saw the 5, 6 & 11, relationship right off (it shows up in the squares we were supposed to know by heart), I did not even look at the problem that way, until you mentioned solving it with "simple math". And by then, Tach's first post.., m < 4 had already put 11^3 = 1331, in my mind and Alpha had already suggested k = 0. So, yes I do think it could have been done in less than ten minutes, but I am not sure this is what you were supposed to be looking for. Have you worked it all the way out yet? And what math class was the problem from?
I didn't know and I looked it up. If you're in an exam then I guess calculator roulette would be my favoured approach. Please Register or Log in to view the hidden image!
The problem was fully resolved by post 9, and post 16 was adequately addressed by post 18, but thanks for the advice anyway. Please Register or Log in to view the hidden image!
its not a problem my teacher set me, I found it going through past IMC papers. You have 90 minutes to do 25 questions, but they're not looking for in-depth analysis, it's actually multiple choice. I'll be sitting this years IMC on the 2nd of february so i want to get some practice. Thanks for the help Please Register or Log in to view the hidden image!
If you are in an exam, seeing patterns like this can be an asset, but it is not required to solve the problem and most of the time not the best way to prepare. Now that you know the "factoid" above, it will likely at some time be of use, and if this problem or one derived from it comes up the answer may just jump out at you. However, the best advice is implied in Pete's post. Learn how to use and apply modular arithmetic. It will serve better for a wider variety of problems. Sometimes patterns as in this problem, can lead to quick solutions, especially in a multiple choice situation. But they are short cuts that often have limited application, apart from recognizing patterns.
Given that it's unique integers, and a multiple-choice test (i.e. no need to show work; easy to check answer), it seems like your best bet in this situation is simple examination (e.g. bounds, etc) with a bit of plug-and-chug.
This is an approach which may wind up being of use, but not within a linear context. What I mean by this, is that if in a multiple choice and timed test, you find yourself reduced to "plug and chug" or guessing, the best approach would be to skip the question and return to it after having addressed those questions you can answer from a better approach. Never waste time guessing or with a "plug and chug" approach! until you have answered all questions you can. Not including achedemic situations, I have taken six or seven different timed Licencing exams, ranging from a couple hours to a full day, and tested high enough, that on at least two I have been listed as a subject matter expert by the state of California. You can add another several pre-employment exams, to that list. This is general advice for a multiple choice exam... If you are allowed to return to questions, the best approach is to go through the questions and answer those you know or know how to find the answer at first reading. Skip the rest and then come back after having answered the "easy" ones. Often many of the questions where the answer was not clear in the first reading, will seem easier on review. Never guess the first time through.., if the exam allows you come back to questions later. And never waste time with a "plug and chug" approach until you have answered the questions you can. If I don't see the answer or how to find the answer in say 30 seconds, I skip the question and come back later.
I agree with the above; I can't stand memorizing boring data (god I hated History), but test-taking is one one my specialties! One of the benefits of only answering those questions to which you know the answer on the first pass is that, occasionally, clues to previous questions are given in later questions. Also, it gives your brain a chance to "churn" on those difficult problems in the background as you march forward. Another extremely useful tactic with multiple choice tests, when one cannot analyze the question properly, is to simply analyze the question-writer. Here's an example: What is the value of pi to 6 decimal places? A) 3.000000 B) 2.718281 C) 3.141592 D) 31.415926 Even if you didn't know what "pi" meant you should be able to deduce its value from these choices. Option B was included to catch those people that recognized and confused e with pi. Option D was included to catch those to whom perhaps the digits, but not the exact value, of pi looked familiar. Option A was included to catch those people that perhaps recalled an obscure history or biblical reference which claimed that pi = 3. Even if we don't know why those particular options were chosen, we can still compare them. Taken together, we can see that pi likely starts with 3, is closer to 3 than 30, and isn't an integer...which leaves a single choice.
