# proof4 .pdf

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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&lt;|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
theorem.
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
numbers.