To find the least common multiple (LCM) of 7 and 12, we can use the method of prime factorization or listing the multiples.
-
Prime factorization:
- The prime factorization of 7 is \( 7^1 \).
- The prime factorization of 12 is \( 2^2 \times 3^1 \).
To find the LCM, take the highest power of each prime number from the factorizations:
- From 7, take \( 7^1 \).
- From 12, take \( 2^2 \) and \( 3^1 \).
Hence, the LCM is: \[ LCM = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 \] \[ = 12 \times 7 = 84 \]
-
Multiples method (optional):
- The multiples of 7 are: \( 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, \ldots \)
- The multiples of 12 are: \( 12, 24, 36, 48, 60, 72, 84, \ldots \)
The first common multiple is 84.
Thus, the least common multiple of 7 and 12 is \( \boxed{84} \).