Finding the Least Common Multiple (LCM) of Numbers

The problem asks us to find the Least Common Multiple (LCM) of three numbers: 37, 111, and 148. The LCM is the smallest positive integer that is a multiple of all the given numbers.

To find the LCM, we can use the prime factorization method. This involves finding the prime factors of each number and then combining them to get the LCM.

Prime Factorization of 37, 111, and 148

Let's find the prime factorization for each number:

• 37: 37 is a prime number. It can only be divided by 1 and itself.
  Prime factorization of 37 is ${37}^1$.
• 111: We can divide 111 by small prime numbers. It is divisible by 3.
  ${111 \div 3 = 37}$.
  Since 37 is a prime number, we stop here.
  Prime factorization of 111 is ${3 \times 37 = 3^1 \times 37^1}$.
• 148: We can divide 148 by small prime numbers. It is an even number, so it's divisible by 2.
  ${148 \div 2 = 74}$.
  74 is also even, so divide by 2 again.
  ${74 \div 2 = 37}$.
  Since 37 is a prime number, we stop here.
  Prime factorization of 148 is ${2 \times 2 \times 37 = 2^2 \times 37^1}$.

Here is a summary of the prime factorizations:

Calculating the LCM using Prime Factors

To find the LCM, we take the highest power of each prime factor that appears in any of the factorizations. The prime factors involved are 2, 3, and 37.

• Highest power of 2: The prime factor 2 appears in the factorization of 148 as ${2^2}$. It doesn't appear in 37 or 111 (which is like ${2^0}$). The highest power is ${2^2}$.
• Highest power of 3: The prime factor 3 appears in the factorization of 111 as ${3^1}$. It doesn't appear in 37 or 148 (which is like ${3^0}$). The highest power is ${3^1}$.
• Highest power of 37: The prime factor 37 appears in all factorizations as ${37^1}$. The highest power is ${37^1}$.

Now, we multiply these highest powers together to get the LCM:

${LCM(37, 111, 148) = (\text{Highest power of 2}) \times (\text{Highest power of 3}) \times (\text{Highest power of 37})}$

${LCM(37, 111, 148) = 2^2 \times 3^1 \times 37^1}$

${LCM(37, 111, 148) = 4 \times 3 \times 37}$

${LCM(37, 111, 148) = 12 \times 37}$

Now, calculate the final product:

${12 \times 37 = 12 \times (30 + 7) = (12 \times 30) + (12 \times 7) = 360 + 84 = 444}$

So, the Least Common Multiple of 37, 111, and 148 is 444.

Revision Table: LCM Calculation Steps

Additional Information on LCM and GCF

Understanding LCM is often paired with understanding the Greatest Common Factor (GCF) or Highest Common Divisor (HCF).

• LCM: The smallest number that is a multiple of all the given numbers. Useful when dealing with problems involving cycles or events repeating at different intervals (e.g., bus schedules, flashing lights).
• GCF/HCF: The largest number that divides into all the given numbers without leaving a remainder. Useful when simplifying fractions or dividing items into equal groups.

For the numbers 37, 111, and 148:

• We found the LCM is 444.
• Let's find the GCF. The common prime factors are those present in all factorizations. Only 37 is common to all three. The lowest power of 37 is ${37^1}$.
  GCF(37, 111, 148) = ${37^1 = 37}$.
  So, the GCF is 37.

There is a relationship between LCM and GCF for two numbers, 'a' and 'b': ${a \times b = LCM(a, b) \times GCF(a, b)}$. This relationship does not generally hold true for three or more numbers directly in the same form.

The final answer is 444.