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Calvin Lin:

Find the number of coprime pairs (a, b) of positive integers subject to 0 100, such that a|(b2-11) and b|(a2-11).

There would be a small correction to the problem. You say, a|b2-11. Then a = [ b^2 -11] / k1. Where k1 is a natural number. Then, [b- 11^(1/2)].[b+11^(1/2)] / k1 = a = prime number. Then either [b- 11^(1/2)] /k1 OR [b+11^(1/2)]/k1. Because a>1 according to the given conditions. But b is a natural number. And 11^(1/2) is not a natural number. Also k1 is a natural number. Then we have a contradition. If you input some square number instead of 11, then the problem may okay.

Note by Geerasee Wijesuriya
3 years ago

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Note that "coprime pairs of positive integers" means that $$\gcd (a,b) = 1$$.

It does not imply that $$a$$ and $$b$$ are prime numbers.

For example, $$(6, 35 )$$ are coprime, but neither of the numbers are prime.

Staff - 3 years ago

Calvin Lin: Actually, a and b are not needed to be prime numbers. According to my previous argument, then a =1. Then according to the problem's conditions b can take only 1, 2,5. Then answer should be 3.

- 3 years ago

Sorry but I do not understand your argument.

Note that $$(5, 14)$$ is also a solution, since $$5^2 - 11 = 14$$ and $$14^2 - 11 = 185$$. This contradicts your claim that $$a = 1$$.

Staff - 3 years ago

No. Actually, according to my previous argument, [b- 11^(1/2)].[b+11^(1/2)] / k1 = a. But, if you put a square number(k^2) instead of 11, then gcd(a, b+k) or gcd(a, b-k) not equals to 1. Then according to the gcd operations, gcd(k,a) =1 and gcd(k,b) =1. (The operation is gcd(a,b) = gcd(a+m.b ,b) ). Therefore, if you choose k^2 instead of 11, such that gcd(k,ab) =1.

- 2 years, 11 months ago