What is the Smallest Number Divisible by 3, 4, and 5?

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When trying to find the smallest number that is divisible by 3, 4, and 5, a fundamental concept in number theory comes into play - the least common multiple (LCM).

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Prime Factorization and LCM Calculation

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To determine the LCM of 3, 4, and 5, let's start with their prime factorization:

" "" "3 31" "4 22" "5 51" "" "

Since the highest power of each prime factor is taken, we can calculate the LCM as follows:

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LCM 22 × 31 × 51 4 × 3 × 5

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This results in LCM 60. Therefore, the smallest number divisible by 3, 4, and 5 is 60.

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Smallest Three-Digit Number Divisible by 3, 4, and 5

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When seeking the smallest three-digit number divisible by 3, 4, and 5, LCM (60) becomes our guide. Multiples of 60 include:

" "" "60" "120" "180" "240" "300" "" "

The smallest three-digit number in this list is 120, making it the answer we seek.

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Verifying the Solution with a Bash Script

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To further validate this, a simple bash script can be written to find the smallest three-digit number that is divisible by 3, 4, and 5.

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#!/bin/basha100while [ $a -lt 1000 ]do  if !((a % 3))  !((a % 4))  !((a % 5))  then    echo $a  fi  let aa 1done

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This script checks each number from 100 to 999, and prints only those that are divisible by 3, 4, and 5. The output confirms that the smallest number is 120, aligning with our solution.

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Conclusion

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Through the use of prime factorization and the LCM concept, we can systematically find the smallest number divisible by 3, 4, and 5. Utilizing a straightforward script, we can further verify the result. The smallest three-digit number that is divisible by 3, 4, and 5 is undoubtedly 120.