To find the least common multiple (LCM) of 80, 96, and 120, we can use the prime factorization method:

1. Prime factorization:
   • \(80 = 2^ 4 \times 5\)
   • \(96 = 2^ 5 \times 3\)
   • \(120 = 2^ 3 \times 3 \times 5\)
2. Identify the highest powers of all prime factors:
   • The highest power of 2 is \(2^ 5\).
   • The highest power of 3 is \(3^ 1\).
   • The highest power of 5 is \(5^ 1\).
3. Calculate the LCM:
   • LCM = \(2^ 5 \times 3^ 1 \times 5^ 1\)
   • LCM = \(32 \times 3 \times 5\)
   • LCM = \(96 \times 5\)
   • LCM = 480

Supplemental Knowledge

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the numbers. Finding the LCM involves several steps:

1. Prime Factorization: Break down each number into its prime factors.
2. Identify Common and Unique Factors: For each prime number that appears in the factorizations, take the highest power of that prime.
3. Multiply These Factors Together: The product of these factors gives you the LCM.
   For example, let's find the LCM of 8 and 12:

• Prime factorization of 8: \(2^ 3\)
• Prime factorization of 12: \(2^ 2 \times 3\)
  The LCM will include \(2^ 3\) (the highest power of 2) and \(3\):
• LCM = \(2^ 3 \times 3 = 24\)

Practical Insights

Imagine planning a party and needing packs of soda cans with different capacities: 80 cans per pack, 96 cans in another, 120 in yet another and the LCM being calculated will enable you to purchase enough packs so as to have equal numbers without leftovers from any type. In such an instance, the lowest common multiple of LCM would give an answer on what should be purchased to meet this criterion.
Learning how to find the LCM can come in handy in numerous real-life applications such as scheduling events, merging frequencies in signal processing applications or managing inventory in businesses.

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