So θ ≡ 0 mod 45, but θ ≢ 0 mod 90. - Redraw
Understanding the Modular Condition: θ ≡ 0 mod 45, but θ ≢ 0 mod 90
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.
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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.
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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:
