Answer :
To determine the least common multiple (LCM) of 10, 12, and 14 using prime factorization, we first need to find the prime factors of each number:
10: 10=2×5
12: 12=2^(2)×3
14: 14=2×7
Next, we identify the highest power of each prime number that appears in the factorizations:
Prime 2: The highest power is
2^(2) (from 12).
Prime 3: The highest power is
3^(1) (from 12).
Prime 5: The highest power is
5^(1) (from 10).
Prime 7: The highest power is
7^(1) (from 14).
The LCM is found by multiplying these highest powers together:
=2^(2)×3^(1)×5^(1)×7^(1)
Calculating this:
LCM=4×3×5×7
=>
LCM=12×5×7
=>
LCM=60×7
=>
LCM=420
So, the least common multiple of 10, 12, and 14 is 420.
