Original filename: proof4.pdf
This PDF 1.7 document has been generated by WPS Office / , and has been sent on pdf-archive.com on 28/08/2017 at 01:39, from IP address 108.218.x.x.
The current document download page has been viewed 163 times.
File size: 69 KB (1 page).
Privacy: public file
Download original PDF file
4. Theorem: Any odd number can be expressed by 4n + 1 or 4n + 3,
where n is an integer.
It is known that ∀a,b∈Z,b≠0:∃!q,r∈Z:a=qb+r,0≤r<|b| can represent
all natural numbers.
Using the given expressions, qb + r can be written as 4n + r.
4 has 4 possible remainders: 0, 1, 2, 3.
4n + 0 expresses all even numbers divisible by 4. Not relevant.
4n + 1 expresses all odd numbers that are 1 greater than 4n. Relevant
and included in the theorem.
4n + 2 expresses all even numbers not divisible by 4. Not relevant.
4n + 3 expresses all other odd numbers. Relevant and included in the
These four expressions can express all natural numbers. Since 4n + 0 and
4n + 2 only express even natural numbers, they are not relevant. The
remaining two expressions, 4n + 1 and 4n + 3, can express all natural odd