Because 18n = 45k → 2n = 5k → n = 5a, k = 2a → θ = 18×5a = 90a → so only multiples of 90. - Crosslake
Why 18n = 45k Implies Only Multiples of 90: A Simple Mathematical Breakdown
Why 18n = 45k Implies Only Multiples of 90: A Simple Mathematical Breakdown
When solving equations involving variables, especially in algebra, understanding the chain of logic can clarify their implications. One such equation—18ₙ = 45ₖ—may seem straightforward, but unpacking it reveals deeper relationships that define specific values. Let’s explore step-by-step why this equation leads uniquely to θ = 90a, meaning only multiples of 90 are valid solutions.
The Equation: 18ₙ = 45ₖ
Understanding the Context
This notation expresses a proportional relationship between two expressions: 18 times the unknown variable n, and 45 times another unknown variable k. At first glance, the equation seems symbolic, but we can simplify it through algebraic manipulation.
Step 1: Simplify 18ₙ = 45ₖ
Although written with superscripts (often used for exponents), here n and k represent variables, so 18ₙ means 18 times n, and 45ₖ means 45 times k. Thus, the equation becomes:
Key Insights
18n = 45k
Step 2: Reduce the Equation
We simplify 18n = 45k by dividing both sides by the greatest common divisor of 18 and 45, which is 9:
(18n ÷ 9) = (45k ÷ 9)
→ 2n = 5k
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This reduced form reveals a direct proportional relationship between n and k. Because 2 and 5 have no common factors, n must increase in multiples of 5 and k in multiples of 2 to maintain the equality.
Step 3: Express One Variable in Terms of a Shared Parameter
From 2n = 5k, solve for n:
n = (5k) / 2
For n to be an integer, k must be even. Let k = 2a (where a is any positive integer). Substituting:
n = (5 × 2a) / 2 = 5a
Step 4: Solve for θ (or θ = 18n)
Earlier, we derived θ = 18n. Substituting n = 5a:
