To find the smallest value of [tex]\( n \)[/tex] such that the least common multiple (LCM) of [tex]\( n \)[/tex] and 15 is 45, follow these steps:
1. Understand the problem: We need to find the smallest integer [tex]\( n \)[/tex] for which the least common multiple of [tex]\( n \)[/tex] and 15 equals 45.
2. LCM definition: Recall that the least common multiple of two numbers is the smallest positive integer that is divisible by both numbers.
3. Prime Factorization:
- The prime factorization of 15 is [tex]\( 15 = 3 \times 5 \)[/tex].
- The prime factorization of 45 is [tex]\( 45 = 3^2 \times 5 \)[/tex].
4. Relationship of LCM:
- The LCM of two integers takes the highest powers of all prime factors that appear in either number.
- For the LCM of [tex]\( n \)[/tex] and 15 to be 45, [tex]\( n \)[/tex] must contribute such that the combined factors result in [tex]\( 3^2 \times 5 \)[/tex].
- [tex]\( n \)[/tex] must therefore include at least one more factor of 3 to make [tex]\( 3^2 \)[/tex].
5. Factors Analysis:
- Since 15 is already [tex]\( 3 \times 5 \)[/tex], [tex]\( n \)[/tex] needs at least an additional 3.
- The simplest higher form, considering the factors we need, would be [tex]\( n \)[/tex] containing another factor of 3.
- As a result, [tex]\( n = 3 \times 3 = 9 \)[/tex].
6. Check:
- Determine if [tex]\(\text{LCM}(9, 15) \)[/tex] actually equals 45.
- The prime factorization of 9 is [tex]\( 3^2 \)[/tex].
- Combining with 15 ([tex]\( 3 \times 5 \)[/tex]), the LCM would include [tex]\( 3^2 \)[/tex] from 9 and the [tex]\( 5 \)[/tex] from 15, resulting in [tex]\( 3^2 \times 5 = 45 \)[/tex].
Thus, the smallest value of [tex]\( n \)[/tex] such that the least common multiple (LCM) of [tex]\( n \)[/tex] and 15 is 45, is [tex]\( n = 9 \)[/tex].
