Thread: Divisible by 16

Prove that if m and n are odd integers, then is divisible by 16.

i think you need to specify a domain for m and n,

for example if m and n are both 1 the rule doesn't work

or if m is 1 and n is -1

What do you mean by "domain" here? Usually, this term is used to specificy the set of arguments for some function.

It was clearly stated that m and n are integer and odd.

In fact the "rule" does work for 1 and -1. Let me make it clear:

The term "divisible by" means no more (or less) than the fact that dividing by, say 16, leaves no remainder. So if m = 1, n =1 in the present case, then 16 divides 0, no remainder.

organic god appears to be labouring under the delusion that 0 is not divisible by 16. Well, it is – as pointed out by Guitarist.

Anyway …

Since and are odd, is odd. Hence

(the square of any odd integer is congruent to 1 modulo 8)

Hence is divisible by 8.

Also, is even (i.e. divisible by 2).

Hence is divisible by 16.

yeh sorry about that guys, obviously 0 does divide by 16...not sure why i thought it didnt.
something that has been bugging me for a while actually and fits in quite nicely...is it possible to classify a complex number as even or odd.

i was thinking about it. and i had some basic ideas.
for simple numbers like i
i^2 is an odd number.
therefore i must be an odd number

and then i thought about numbers like ai where a is some interger.

and so if (ai)^2 is odd
then a must be odd also be odd.
and so ai can be classed as odd.

but unfortunately my knowledge of complex numbers is not deep so i am not sure if you can classify complex numbers as odd or even.

especially when you get a complex number which is a mixture of real and imaginary parts, how could you classify that.

I'm not sure how far you could really stretch that, since, by the same reasoning, would be even. Even restricted to just complex integers (is that the right word?) I'm not sure even and odd really makes all that much sense.

What you'd have to do is lay out your definitions in a very clear way. From there, it shouldn't be too hard to work out whether something is even or odd or if the question even makes sense.

I have never heard of the terms “even” and “odd” applied to any numbers other than the integers.

If you don’t stick to integers, you might end up with absurd results. Consider the fact that the product of two odd integers is odd. If you try and “generalize” to non-integers, you would get, for example, – and are you then going to say that 2 is odd?!?!

yeh that was a problem i kept having in my own reasoning and so i didnt think you could say complex numbers are odd or even.

Odd or even only applies to intergers then i suppose