What is the least common multiple (LCM) of the numbers from 1 to 10?

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The least common multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by each of the numbers in that set. To find the LCM of the numbers from 1 to 10, we can follow a systematic approach.

The numbers in this range are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Step-by-Step Calculation

  1. List the prime factors: To find the LCM, we first need the prime factorization of each number.
    • 1: (not used, as every number is divisible by 1)
    • 2: 2
    • 3: 3
    • 4: 2 x 2 (or 22)
    • 5: 5
    • 6: 2 x 3
    • 7: 7
    • 8: 2 x 2 x 2 (or 23)
    • 9: 3 x 3 (or 32)
    • 10: 2 x 5
  2. Identify the highest powers of each prime factor: We’ll take the highest powers of all prime factors present in the numbers:
    • 2: 23 (from 8)
    • 3: 32 (from 9)
    • 5: 51 (from 5 and 10)
    • 7: 71 (from 7)
  3. Calculate the LCM: Now, multiply these highest powers together:
    4. LCM = 23 x 32 x 51 x 71
  4. Compute the result:
    5. LCM = 8 x 9 x 5 x 7

    First, calculate:

    8 x 9 = 72
    72 x 5 = 360
    360 x 7 = 2520

Thus, the least common multiple (LCM) of the numbers from 1 to 10 is 2520.

Conclusion

Understanding how to find the LCM is useful in various applications such as problem-solving in mathematics, scheduling, and even in programming algorithms. The LCM of 1 to 10 being 2520 illustrates how numbers interact and helps in simplifying complex computations.

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