21cracked the first one...answer comes to be 1328
1) Find the number of positive integers n for which n≤1991 and 6 is a factor of (n2+3n+2)
2)determine all pairs (m,n) of positive integers for which 2m +3n is a perfects square
3)find the remainder when 1992 is divided by 92
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6 Answers
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1708$This Can Also be Solved by Using \underline{\underline{Binomial- Theorem}}.
1708$\underline{\underline{Using Binomial Theorem}}\Rightarrow\\\\ $92=23\times4$ and $19=23-4$\\\\ So $19^{92}=(23-4)^{92}=23^{92}+4^{92}+92K$\\\\ in that $23^{92}$ gives remainder $69$, by $\frac{{23}^{92}}{23\times4}$\\\\ and $4^{92}$ gives remainder $72$, by $\frac{4^{92}}{23\times4}$\\\\ hence remainder is \boxed{\boxed{$\frac{69+72}{92}=49}}$
3412) http://www.goiit.com/posts/list/algebra-with-proof-determine-the-no-of-pairs-m-n-so-that-3-m-76580.htm