The least common multiple (LCM) of 9 and 13 is 117. Think about it: this means 117 is the smallest number that both 9 and 13 can divide into without leaving a remainder. Understanding how to calculate the LCM is essential in mathematics, particularly when solving problems involving fractions, ratios, or real-world scenarios like scheduling.
To determine the LCM of two numbers, there are three primary methods: prime factorization, listing multiples, and using the greatest common divisor (GCD). Each approach provides a systematic way to arrive at the correct answer, and they all confirm that the LCM of 9 and 13 is 117.
Some disagree here. Fair enough.
Prime Factorization Method
The prime factorization method involves breaking down each number into its prime factors and then multiplying the highest powers of all primes involved. For 9, the prime factors are 3 × 3 (or 3²). For 13, since it is a prime number, its only prime factor is 13 itself. To find the LCM, we take the highest power of each prime number present in either factorization. This gives us 3² (which is 9) and 13. Multiplying these together:
9 × 13 = 117
This method is efficient because it directly identifies the building blocks of each number, ensuring no smaller common multiple exists.
Listing Multiples Method
Another straightforward approach is to list the multiples of each number until a common one is found. Starting with 9, the multiples are:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, ...
For 13, the multiples are:
**13, 26, 39, 52, 65, 78, 91, 104, 117
