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07 Dec 2019, 16:58
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I understand that z and z+1 share no common factors (other than 1), because they are 1 number apart and, thus, only 1 could divide both.
But I cannot understand why z and k*z+1 cannot have any common factors (other than 1).
This is a method Bunuel used on this question: if-x-and-y-are-positive-integers-such-that-x-8y-12-what-is-the-126743.html
Thanks. Best,
But I cannot understand why z and k*z+1 cannot have any common factors (other than 1).
This is a method Bunuel used on this question: if-x-and-y-are-positive-integers-such-that-x-8y-12-what-is-the-126743.html
Thanks. Best,
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07 Dec 2019, 19:38
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sweetlyimproved
If k and z are integers, k*z will come in the table of z and all factors of z will be there in k*z.
Now k*z and k*z+1 are consecutive integers and will not have any common factors.
But all factors of z are in k*z, so z and k*z+1 will also not have any common factors
10 Dec 2019, 18:18
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sweetlyimproved
Start by assuming that you're wrong: what if z and kz + 1 share a factor? For example, what if 2 is a factor of both of them?
Then z = 2x. (You don't know what x is, but it's definitely an integer, because 2 is a factor of z.) So, kz + 1 = 2kx + 1.
But, 2 definitely isn't a factor of 2kx + 1, since it's an odd number.
Or what if 3 is a factor of both of them? Then z = 3x, and kz + 1 = 3kx + 1. But, 3 can't be a factor of 3kx + 1, because that number is 1 greater than a multiple of 3.
Make sense?
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
