The LCM (Least Common Multiple) of 7 and 12 is 84. The LCM of any two integers is the value that is evenly divisible by the two values. The smallest number among all frequent multiples of 7 and 12 is the LCM of 7 and 12. To find the LCM of 7 and 12, there are three typical methods: listing multiples, division method, and prime factorization. Here are the steps to find the LCM of 7 and 12 using each method:

• Listing multiples: List the multiples of 7 and 12 until you find the smallest number that is common to both lists. The first few multiples of 7 and 12 are (7, 14, 21, 28... , . . . ) and (12, 24, 36, 48, 60, 72, 84, . . . ) respectively. The common multiples from the multiples of 7 and 12 are 84, 168, . . . The smallest common multiple is 84.

• Division method: Divide the numbers (7, 12) by their prime factors to get the LCM of 7 and 12 using the division method (preferably common). The LCM of 7 and 12 is calculated by multiplying these divisors. The prime factorisation of 7 and 12, respectively, is given by: 12 = (2 × 2 × 3) = 22 × 31 7 = 7¹. No further division can be done. Hence, LCM (7, 12) = 84.

• Prime factorization: Find the prime factorization of each number and multiply the highest power of each prime factor together. The prime factorisation of 7 and 12, respectively, is given by: 12 = (2 × 2 × 3) = 22 × 31 7 = 7¹. The LCM of 7 and 12 is the product of all prime numbers on the left, i.e. LCM(7, 12) by division method = 2 × 2 × 3 × 7 = 84. Hence, the LCM of 7 and 12 by prime factorization is 84.

Therefore, the LCM of 7 and 12 is 84.