Find the Lowest Common Multiple (LCM) of 108 and 120 - Edexcel - GCSE Maths - Question 2 - 2019 - Paper 1

To find the LCM of two numbers, we can use the prime factorization method. First, let's find the prime factors for both 108 and 120.

1. Prime Factorization of 108:

   • 108 can be divided by 2:

   $108 = 2 \times 54$

   • 54 can again be divided by 2:

   $54 = 2 \times 27$

   • Next, 27 is divided by 3:

   $27 = 3 \times 9$

   • Finally, 9 can be factored into:

   $9 = 3 \times 3$

   Therefore, the prime factorization of 108 is:

   $108 = 2^2 \times 3^3$

2. Prime Factorization of 120:

   • 120 can be divided by 2:

   $120 = 2 \times 60$

   • 60 can be divided by 2:

   $60 = 2 \times 30$

   • 30 can be divided by 2 again:

   $30 = 2 \times 15$

   • Finally, 15 can be divided into:

   $15 = 3 \times 5$

   Therefore, the prime factorization of 120 is:

   $120 = 2^3 \times 3^1 \times 5^1$

3. Finding the LCM:

   • The LCM is found by taking the highest power of each prime factor from both factorizations:
   • From 108: $2^2$ and $3^3$
   • From 120: $2^3$, $3^1$, and $5^1$
   • The highest powers are:
     • $2^3$
     • $3^3$
     • $5^1$
   • Therefore, the LCM is:

   $LCM = 2^3 \times 3^3 \times 5^1$

4. Calculating the LCM:

   • Finally, calculate it:

   $LCM = 8 \times 27 \times 5$

   • $8 \times 27 = 216$, and then $216 \times 5 = 1080$.

Thus, the Lowest Common Multiple of 108 and 120 is 1080.