find number ab: ab — это число, где a+b=ab-27 и a+b=a^2+b^2-76. Найдите наибольшее число ab.

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politexnik

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ab — это число, где a+b=ab-27 и a+b=a^2+b^2-76. Найдите наибольшее число ab. Ответ 85. Нужно решить, не вставляя цифр так, чтобы они подходили.
 
politexnik said:
ab — это число, где a+b=ab-27 и a+b=a^2+b^2-76. Найдите наибольшее число ab. Ответ 85. Нужно решить, не вставляя цифр так, чтобы они подходили.
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Google translation...
"ab is the number where a+b=ab-27 and a+b=a2+b2-76. Find the largest number ab.
Answer 85.
It is necessary to solve without inserting numbers so that they fit."

Where is your attempt to solve this problem? (Где ваша попытка решить эту проблему?)

Now you are posting your problems without even bothering to translate them into English!
(Теперь вы публикуете свои задачи, даже не удосужившись перевести их на английский язык!)

It has already been suggested that you stick to other forums where answers will be provided with no input fom you. That is not how this forum works! Please obey the rules of this forum if you wish to post here.
(Вам уже предлагали придерживаться других форумов, где ответы будут предоставлены без вашего участия. Форум так не работает! Пожалуйста, соблюдайте правила этого форума, если вы хотите написать здесь)

It is perhaps time that this member was simply removed from the forum as he or she continues to refuse to abide by the forum's rules. ?
(Возможно, пришло время просто удалить этого участника с форума, поскольку он или она продолжает отказываться соблюдать правила форума. ?)
 
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politexnik said:
ab — это число, где a+b=ab-27 и a+b=a^2+b^2-76. Найдите наибольшее число ab. Ответ 85. Нужно решить, не вставляя цифр так, чтобы они подходили.
Click to expand...
The answer is not 85.
 
politexnik said:
ab — это число, где a+b=ab-27 и a+b=a^2+b^2-76. Найдите наибольшее число ab. Ответ 85. Нужно решить, не вставляя цифр так, чтобы они подходили.
Click to expand...
Is this site a dumping ground for your homework, which you don't even bother to translate? If you just need solutions, learn to use AI tools.
 
He probably didn't read the forum rules here, just like he apparently didn't at Math Forums, either.

-Dan
 
Odd problem.

It is soluble by algebra and arithmetic, but it is quite ugly (I used the rational root theorem to solve the implied quartic). The solution set of a and b is four distinct real pairs. I wonder if the original problem specified integer solutions. I am not sure what the problem is supposed to teach other than that systems of non-linear equations are not fun.

Did anyone find an elegant way (e.g., substitution of variables) to solve this other than brute force?
 
BigBeachBanana said:
The answer is not 85.
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Don't know whether 85 is the largest number that works, but it does satisfy both equations. Just need to keep in mind that the first "ab" is a 2-digit number consisting of digits a and b, and the second "ab" (in a+b=ab-27) is a product.
 
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Since he's not responding here, I'll let y'all know that the "ab" in the problem is actually 10a + b.

-Dan
 
JeffM said:
Odd problem.

It is soluble by algebra and arithmetic, but it is quite ugly (I used the rational root theorem to solve the implied quartic). The solution set of a and b is four distinct real pairs. I wonder if the original problem specified integer solutions. I am not sure what the problem is supposed to teach other than that systems of non-linear equations are not fun.

Did anyone find an elegant way (e.g., substitution of variables) to solve this other than brute force?
Click to expand...
I initially found the quartic, and then just used WolframAlpha to check the answer, but just now I found a nice way. Manipulating a+b and ab, I found that a+b = 13 or -10, from which we get the answers: 5 and 8, or [imath]-5\pm2\sqrt{2}[/imath].

But now I see that others have reinterpreted the question to give (perhaps) the "correct" answer, so this is unnecessary.

No, I take that back: a and b are 8 and 5, so that gives "85".

I imagine that in Russian it is a little clearer that they are saying "Find ab, in the sense of the number formed from digits a and b".
 
Dr.Peterson said:
I imagine that in Russian it is a little clearer that they are saying "Find ab, in the sense of the number formed from digits a and b"
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I tried searching for the problem statement in Russian, no luck. I'm guessing the first ab used a different font or it was obvious from the context that it was not a product.
 
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