Odd power odd

[AntonioLao]

If square of a number is odd then the number itself is odd. If cube of a number is odd then the number is odd. If the 4th power of a number is odd then the number is odd. If the 5th power of a number is odd then the number is odd. If the 6th power of a number is odd then the number is odd. If the 7th power of a number is odd then the number is odd. If the 8th power of a number is odd then the number is odd.

A proof for the 2nd power is given as the following: Let X be an odd integer then X is 1 more or 1 less than an even integer. So X=1+2Y or X=2Y-1 for some integer Y. Therefore X=(1+2Y) or X=(2Y-1). Direct expansions give X=1+4Y+4Y or X=4Y-4Y+1. Factoring the 4’s give X=1+4(Y+Y) or X=1+4(Y-Y), since 4(Y+Y) and 4(Y-Y) are both even for any integer Y, therefore 1+4(Y+Y) and 1+4(Y-Y) is odd and likewise X is odd.

[Cubola Zaruka]

If a^b is not a multiple of c, a is not a multiple of c. If a^b is a multiple of c, a is a multiple or c or a^d=c. This is because all numbers have a unique prime factorisation. This can be extended to prove that any nth power root of a integer is another integer or (exclusive) irrational because, if e^f/g=h/i, both h and i have to be multiples of e, so the fraction would end up being simplifiable an infinite number of times.