Consistency (Meta) Math Example 4

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Example 4

medium
Determine whether the conditions 'nn is a prime number' and 'nn is divisible by 4' are consistent. If so, find an example; if not, explain why.

Solution

  1. 1
    A prime number has exactly two divisors: 1 and itself.
  2. 2
    If nn is divisible by 4, then 4∣n4 \mid n, meaning nn has at least the divisors 1,2,4,n1, 2, 4, n.
  3. 3
    For n>4n > 4, this gives more than two divisors, contradicting primality. For n=4n=4: 44 is not prime.
  4. 4
    The two conditions are inconsistent — no prime is divisible by 4.

Answer

Inconsistent — no prime number is divisible by 4\text{Inconsistent — no prime number is divisible by 4}
Testing the consistency of two properties by checking whether any number could satisfy both reveals a genuine constraint. Divisibility by 4 and primality are mutually exclusive.

About Consistency (Meta)

The property of a set of mathematical statements having no internal contradictions — all statements can be simultaneously true within the same system.

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More Consistency (Meta) Examples

Example 1 easy

A student claims: 'The set [formula].' Check whether this definition is consistent.

Example 2 medium

Check whether the system of equations [formula] and [formula] is consistent.

Example 3 easy

A proof assumes both '[formula] is even' and '[formula] is odd'. Is this assumption consistent? What

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