Let n be a positive odd integer. Prove that n4 ≡ 1 (mod 16). You may use any of the mod rules provided in class (In addition, if you end up getting an expression of the form a (3a+1) somewhere in your work where a is an integer, separately end up proving that this expression is always even and use that result to complete your proof).

If is odd, then we can write for some .

Expanding gives

This reduces to

Notice that is 8 times the product of two consecutive integers, one even and the other odd. This product is even and thus divisible by 2, so that where , and the equivalence further reduces to

as required.

answer: 19 is   2

step-by-step explanation: i just looked at your answer and wrote it down

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.