What would be the smallest natural number which when divided either by $$20$$ or by $$42$$ or by $$76$$ leaves a remainder of $$7$$ in each case?

What would be the smallest natural number which when divided either by $$20$$ or by $$42$$ or by $$76$$ leaves a remainder of $$7$$ in each case?

In a quadratic function, the value of the product of the roots $$\left( {\alpha ,\beta } \right)$$ is $$4.$$ Find the value of $$${{{\alpha ^n} + {\beta ^n}} \over {{\alpha ^{ - n}} + {\beta ^{ - n}}}}$$$

Among $$150$$ faculty members in an institute, $$55$$ are connected with each other through Facebook^® and $$85$$ are connected through WhatsApp^®. $$30$$ faculty members do not have Facebook^® or whatsApp^® accounts. The number of faculty members connected only through Facebook^® accounts is ______________.