7Dr. Emily Carter’s Discovery: Maximizing Display Cases with Equal Fossil Spore Samples from All Sediment Layers

In a groundbreaking study, paleontologist Dr. Emily Carter identified fossil spores in three distinct sediment layers from a key geological site: 18 samples from the upper layer, 42 from the middle layer, and 60 from the lower layer. These samples offer invaluable clues about ancient ecosystems and environmental changes across time. Now, Dr. Carter seeks to create equal, informative display cases for public engagement and research — each containing the same number of fossil spores from every sediment layer, with no samples left over. The challenge: what is the maximum number of display cases she can construct under these conditions?

Understanding the Context

Understanding the Problem

To ensure each display case has identical representation from all three layers, the number of samples from each layer per case must evenly divide the total counts: 18 upper, 42 middle, and 60 lower fossils. Thus, the critical question becomes: What is the largest number of cases such that each layer’s samples are fully and equally distributed?

This requires finding the greatest common divisor (GCD) of the three sample counts: 18, 42, and 60.

7Dr. Emily Carter identified fossil spores in three sediment layers: upper (18 samples), middle (42), lower (60). She wants to organize them into display cases such that each case has the same number of samples from each layer, with no samples left. What is the maximum number of display cases she can use?

Key Insights

Finding the Greatest Common Divisor (GCD)

We compute the GCD to determine the maximum number of display cases possible with no leftover samples:

  • Prime factorization:
    • 18 = 2 × 3²
    • 42 = 2 × 3 × 7
    • 60 = 2² × 3 × 5

The common prime factors are 2 and 3, each to the lowest power present:

  • Minimum power of 2 = ²¹ → 2¹
  • Minimum power of 3 = ³¹ → 3¹

Thus:
GCD(18, 42, 60) = 2 × 3 = 6

Continue Reading

Let \( x = \sin \theta \), so:
2x^2 + x - 1 = 0
x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4}

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Final Thoughts

Organizing the Display Cases

Dr. Carter can therefore prepare 6 display cases — the maximum number possible — each containing:

  • 18 ÷ 6 = 3 fossil spore samples from the upper layer
  • 42 ÷ 6 = 7 fossil spore samples from the middle layer
  • 60 ÷ 6 = 10 fossil spore samples from the lower layer

This balanced arrangement ensures each case is rich, consistent, and scientifically meaningful.

Why This Matters

By aligning her display strategy with mathematical precision, Dr. Carter not only honors scientific rigor but also enhances educational storytelling. Using the GCD to balance sample distribution ensures that each case delivers equal scientific value — a hallmark of thoughtful curation in paleontology.

Conclusion

The maximum number of display cases Dr. Emily Carter can create — with uniform, non-empty representation of fossil spores from all three sediment layers — is 6. This achievement reflects both scientific ingenuity and practical display planning, setting a powerful example for interdisciplinary research communication.