Understanding the Modular Condition: θ ≡ 0 mod 45, but θ ≢ 0 mod 90

When exploring modular arithmetic, one frequently encounters conditions that restrict integers into precise equivalence classes. A particularly interesting case arises with the condition:

> θ ≡ 0 mod 45, but θ ≢ 0 mod 90.

Understanding the Context

This means that θ is a multiple of 45, but it is not a multiple of 90. Such a condition defines a specific subset of integers with unique properties, useful in number theory, cryptography, and algorithmic design. In this article, we’ll unpack what this condition means, explore examples, and explore its mathematical implications.

What Does θ ≡ 0 mod 45 Mean?

The expression θ ≡ 0 mod 45 means that θ is divisible by 45. Mathematically, this is written as:
θ = 45k,
where k is any integer.

So θ ≡ 0 mod 45, but θ ≢ 0 mod 90.

Key Insights

In other words, θ lies on an arithmetic sequence with step 45 — all multiples of 45:
..., -90, -135, -180, -225, 0, 45, 90, 135, 180, ...

What Does θ ≢ 0 mod 90 Imply?

The condition θ ≢ 0 mod 90 means that θ is not divisible by 90. This eliminates values in the 90, 180, -90, -180, ... multiples.

So, eliminating these multiples means θ is divisible by 45 but falls strictly between multiples of 90 — in effect, selecting every odd multiple of 45.

Continue Reading

List them:
\( a=0, b=0 \): 1
\( a=1, b=0 \): 3

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Final Thoughts

Specifically, such θ can be written as:
θ = 45(2m + 1) = 90m + 45,
where m is any integer.

These values alternate between 45, -45, 135, -135, etc., skipping every 90.

Numerical Examples

Let’s list some values satisfying θ ≡ 0 mod 45 but θ ≢ 0 mod 90:

  • θ = 45:
    45 ÷ 45 = 1 → OK (multiple of 45)
    45 ÷ 90 = 0.5 → Not an integer → Not divisible by 90

  • θ = -45:
    -45 ÷ 45 = -1 → OK
    -45 ÷ 90 = -0.5 → Not divisible by 90

  • θ = 135:
    135 ÷ 45 = 3 → OK
    135 ÷ 90 = 1.5 → Not divisible by 90

  • θ = -135:
    -135 ÷ 45 = -3 → OK
    -135 ÷ 90 = -1.5 → Not divisible by 90

Now, values that don’t satisfy the second condition (i.e., θ ≡ 0 mod 90) include: