Because 18n = 45k → 2n = 5k → n = 5a, k = 2a → θ = 18×5a = 90a → so only multiples of 90.

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

🔗 Related Articles You Might Like:

📰 Madoka’s Untold Secrets Revealed—Every Flicker Holds the Key to Change! 📰 Madhouse Animes That’ll Make You Hairstand – You Won’t Believe These Masterpieces! 📰 "800-Million Bad Decisions: Why Madhouse Animes Dominate the Genres! 📰 A Ladder 13 Meters Long Leans Against A Wall Reaching Height 12 Meters How Far Is The Base From The Wall 📰 A Laptop Depreciates In Value By 20 Each Year If Its Initial Price Was 1500 What Is Its Value After 3 Years 📰 A Loan Of 15000 Is To Be Repaid In 5 Equal Annual Installments With An Interest Rate Of 6 Per Annum What Is The Amount Of Each Installment 📰 A Mining Engineer Is Designing A Cylindrical Shaft With A Diameter Of 6 Meters And A Depth Of 120 Meters If The Rock Density Is 28 Metric Tons Per Cubic Meter How Many Metric Tons Of Rock Must Be Removed To Create The Shaft 📰 A Mixture Contains 20 Alcohol And The Rest Water If 10 Liters Of Water Are Added To 30 Liters Of This Mixture What Is The New Percentage Of Alcohol 📰 A Mixture Contains 30 Alcohol And 70 Water If 5 Liters Of Pure Alcohol Are Added To 20 Liters Of The Mixture What Is The New Percentage Of Alcohol In The Mixture 📰 A Quadratic Equation Ax2 Bx C 0 Has Roots 3 And 5 If A 2 What Is The Value Of B 📰 A Rectangle Has A Length That Is 3 Times Its Width If The Perimeter Is 96 Meters What Is The Width 📰 A Rectangle Has A Length That Is 3 Times Its Width If The Perimeter Of The Rectangle Is 64 Meters What Is The Width Of The Rectangle 📰 A Rectangles Length Is Twice Its Width If The Perimeter Is 48 Cm What Are The Rectangles Dimensions 📰 A Rectangular Garden Has A Perimeter Of 60 Meters And A Length 5 Meters More Than Twice Its Width Find The Dimensions 📰 A Rectangular Prism Has Dimensions 4 Cm 5 Cm And 6 Cm If Each Dimension Is Increased By 50 What Is The New Volume 📰 A Rectangular Swimming Pool Measures 25 Meters In Length 10 Meters In Width And 2 Meters In Depth If It Is Filled With Water Calculate The Weight Of The Water In Kilograms Density Of Water 1000 Kgm 📰 A Research Team Studying Extremophiles In Hydrothermal Vents Discovers A Microbe That Doubles Every 15 Hours If A Sample Starts With 350 Cells How Many Cells Are Present After 9 Hours Assuming Unrestricted Growth 📰 A Researcher Is Studying The Growth Of A Bacterial Culture The Initial Population Is 500 Bacteria And It Doubles Every 3 Hours How Many Bacteria Will There Be After 9 Hours

Final Thoughts

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: