To calculate the least common multiple (LCM) and greatest common divisor (GCD) of 48 and 72, we need to delve into some basic number theory concepts. The LCM is the smallest positive integer that is divisible by both numbers, while the GCD is the largest positive integer that divides both numbers without leaving a remainder.

Understanding LCM and GCD

The LCM of two numbers can be found using their prime factorizations. For 48, the prime factorization is 2^4 3^1, and for 72, it is 2^3 3^2. To determine the LCM, take the highest power of each prime present in the factorizations: LCM = 2^4 3^2 = 144.

Calculating the GCD

The GCD is determined by taking the lowest power of each prime present in both factorizations. For 48 and 72, the common primes are 2 and 3. Thus, GCD = 2^3 3^1 = 24.

Conclusion

The LCM of 48 and 72 is 144, while the GCD is 24. These calculations help in understanding the relationship between numbers and are useful in various mathematical and real-life applications.