Question: A museum curator is cataloging early computing devices and notes that the number of components in a mechanical computer from 1850 is a three-digit number divisible by both 12 and 15. What is the smallest such number that ends in 0? - Coaching Toolbox
The Smallest Three-Digit Number Over 100 Ending in 0, Divisible by Both 12 and 15
The Smallest Three-Digit Number Over 100 Ending in 0, Divisible by Both 12 and 15
When a museum curator is cataloging early computing devices, identifying historical computing components often involves studying the mechanical components used in 19th-century machines. A particular mechanical computer from 1850 features a three-digit number of parts described as divisible by both 12 and 15 — and crucially, the number ends in 0. What is the smallest such number?
To find this number, we begin by analyzing the divisibility requirements.
Understanding the Context
Step 1: LCM of 12 and 15
Since the number must be divisible by both 12 and 15, it must be divisible by their least common multiple (LCM).
Prime factorizations:
- 12 = 2² × 3
- 15 = 3 × 5
LCM = 2² × 3 × 5 = 60
Thus, the number must be divisible by 60.
Step 2: Restrict to Three-Digit Numbers Ending in 0
We seek the smallest three-digit multiple of 60 ending in 0.
Note: A number divisible by 10 ends in 0. Since 60 is already divisible by 10, all multiples of 60 end in 0. Therefore, we only need the smallest three-digit multiple of 60.
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Key Insights
Step 3: Find the smallest three-digit multiple of 60
The smallest three-digit number is 100. Divide 100 by 60:
100 ÷ 60 ≈ 1.67
The next whole multiple is 2, so:
2 × 60 = 120
120 is a three-digit number, divisible by both 12 and 15, and ends in 0.
Conclusion
Thus, the smallest three-digit number ending in 0 and divisible by both 12 and 15 is 120. This number could plausibly represent the count of fundamental components in an early computing device of 1850, reflecting both the engineering precision and historical accuracy expected in museum curation.
Keywords: museum curator, mechanical computer, 1850 computing device, early computing components, three-digit number, divisible by 12 and 15, ends in 0, LCM 60, historical computing, component count, historical math, museum cataloging
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