Let the coefficients of third, fourth and fifth terms in the expansion of $${\left( {x + {a \over {{x^2}}}} \right)^n},x \ne 0$$, be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to ___________.

If (2021)³⁷⁶² is divided by 17, then the remainder is __________.

Let n be a positive integer. Let

$$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^k} + {{\left( {{{31} \over {32}}} \right)}^k}} \right]} $$. If

$$63A = 1 - {1 \over {{2^{30}}}}$$, then n is equal to _____________.

Let m, n$$\in$$N and gcd (2, n) = 1. If $$30\left( {\matrix{ {30} \cr 0 \cr } } \right) + 29\left( {\matrix{ {30} \cr 1 \cr } } \right) + ...... + 2\left( {\matrix{ {30} \cr {28} \cr } } \right) + 1\left( {\matrix{ {30} \cr {29} \cr } } \right) = n{.2^m}$$, then n + m is equal to __________.

(Here $$\left( {\matrix{ n \cr k \cr } } \right) = {}^n{C_k}$$)