Whole part of a fraction and equations in whole numbers

This is an advanced topic: in the Russian programme it is in the advanced level of Grade 8 (whole part of an algebraic fraction, remainders of powers, equations in whole numbers). It can be skipped in an ordinary lesson.

To take the whole part divide the top by the bottom: n + 3n + 1 = 1 + 2n + 1. The fraction is whole only if n + 1 divides 2.

Last digits and remainders of powers repeat in a cycle. The powers of 7 end in 7, 9, 3, 1, then 7, 9, 3, 1 again. Divide a big index by the length of the cycle and look at the remainder.

For equations in whole numbers use factoring and trying values. If the sum and product of two numbers are given, look for the pair.

Fill in the tables and look for the pattern; in the last task choose x and the table checks the condition.