lcm of 63 and 27
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 63 and 27. To find the LCM, we can use the prime factorization of both numbers, or use the relation between LCM and GCD.
Answer
The LCM of 63 and 27 is $189$.
Answer for screen readers
The least common multiple (LCM) of 63 and 27 is 189.
Steps to Solve
- Find the Prime Factorization of 63
First, we find the prime factors of 63.
Starting with the smallest prime, 2, we see that 63 is odd, so we try 3: $$ 63 \div 3 = 21 $$ Next, factor 21: $$ 21 \div 3 = 7 $$ Since 7 is a prime number, the prime factorization of 63 is: $$ 63 = 3^2 \times 7^1 $$
- Find the Prime Factorization of 27
Next, we find the prime factors of 27.
Starting with the smallest prime, we use 3: $$ 27 \div 3 = 9 $$ Then factor 9: $$ 9 \div 3 = 3 $$ Thus, the prime factorization of 27 is: $$ 27 = 3^3 $$
- Determine the LCM Using Prime Factorizations
To find the LCM, we take the highest powers of each prime factor found in both factorizations.
The prime factors are 3 and 7:
- For 3, the highest power is $3^3$ (from 27)
- For 7, the highest power is $7^1$ (from 63)
Hence, the LCM is: $$ LCM = 3^3 \times 7^1 $$
- Calculate the LCM
Now we calculate the LCM: $$ LCM = 27 \times 7 = 189 $$
The least common multiple (LCM) of 63 and 27 is 189.
More Information
The LCM is the smallest number that is a multiple of both numbers. In practical situations, LCM can help in finding common intervals or synchronous events.
Tips
- Forgetting to use the highest power of primes when calculating the LCM. Always ensure to check each prime factor from both numbers.
- Confusing LCM with GCD (greatest common divisor). Remember that LCM finds the smallest common multiple, while GCD finds the largest common factor.
