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? - Sourci

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.

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.

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Final Thoughts

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