I'm don't know from where to start in this problem. I have to prove that     $(a+b,a^2-ab+b^2)=1 \text{ or } 3$   knowing that $(a,b) = 1$.  I've tried using the method they taught us on class, so I've written that:     $d\ |\ a+b \Rightarrow d\ | a^2 + 2ab + b^2$  and $d\ |\ a^2-ab+b^2 $   $\Rightarrow d\ | \ 3ab$   But from here, I dont know where to go. Usually, I should be able to  find that d divides something in terms of a, and something in terms of b, and therefore, using that $(a:b)=1$ I would get the solution. But this is not the case. Any operation I've tried between the two terms, gives me an other term, but always in terms of a and b, and no a or b, so I'm not able to prove what I'm asked.  Any ideas?
Hint:  $a^2 -ab +b^2 = (a+b)^2 -3ab.$  I know we can say that there exists an $x,y$ such that $ax + by = 1$.  So in this case, $(a+b)x + ((a+b)^2 -3ab)y =1.$  I thought setting $x = (a+b)$ and $y = -1$ would help but that gives me $3ab =1.$  Any suggestions?
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First off, some data: According to COCA word-for-word has 60 usages, 3 of them are &quot;word-for-word translation&quot;. Word-by-word has 26 usages, none of them are &quot;word-by-word translation&quot; (but some with &quot;transcription&quot;). The definition of word-for-word:  : In exactly the same or, when translated, exactly equivalent words : being in or following the exact words, verbatim : one word at a time, without regard for the sense of the whole  Only the last dictionary contains a definition for word-by-word, too:  : one word at a time  The definitions given by The Free Dictionary are, obviously, identical to each other. Google hits: Word-for-word ~21m Word-for-word translation ~318k Word-by-word ~3.8m Word-by-word translation ~95k According to usages and dictionaries word-by-word is, at least, less popular. And assuming that there may be a lot of usages from non-natives among the Google hits, this could be an indicator for word-by-word being even utterly wrong. In another forum I found the following :  When I translate something &quot;literally,&quot; (wörtlich) it still follows the main rules of the language I'm translating into. What you mean is &quot;word-by-word&quot; (wortwörtlich) to me.  I assume that this was written by a German but I don't know it. However, if this would be true a &quot;word-by-word translation&quot; would be a translation where I keep, for instance, the order of the words, disregarding if it makes sense in the target language. Some examples:  Original: word-by-word Word-by-word translation: Wort bei Wort (That's a terrible translation!) Original: It is critical to know... Word-by-word translation: Es ist kritisch zu wissen... (That's a terrible translation!) Original: Ich glaub, ich spinne. Word-by-word translation: I think I spider. (I guess only Germans understand this.)  A &quot;word-for-word translation&quot;, however, would be an attempt to keep the word-choice as close as possible but following the rules of the target language (e.g. order of words) and also considering if the statement still makes sense in the other language. Here are better translations for the examples above:  Wort für Wort Es ist wichtig zu wissen... I think, I'm going nuts. (Actually, this is not a word-for-word translation but rather a .)  So, my questions again:  As neither Oxford nor Merriam-Webster have any entries for word-by-word in their dictionaries: is word-by-word actually valid? If yes, is there any difference between &quot;word-by-word translation&quot; and &quot;word-for-word translation&quot;? If yes again, what is it specifically? 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