Show that there are at least two elements in any subset with three elements of set $$A=\{x\in\mathbb{N}|x=a^4+a^2+1,a\in\mathbb{N}\}$$ whose difference is divisible by 10. I manage to see that if I take two elements of A, say x and y, than $$x-y=a^4+a^2-b^4-b^2=(a-b)(a+b)(a^2+b^2+1)$$ and from here I can't fin anything. thx
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1Hint: If you list the first few values of $x$ you will see a pattern in the last digit. – 3rdMoment Nov 10 '22 at 20:13
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1Hint: look at the values of $a^4+a^2+1 \bmod 10$. – Peter Phipps Nov 10 '22 at 20:14
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thx, I got it! the last digit it is always 1 or 3. from there it is easy. – Numbers Nov 10 '22 at 20:24