To find the Least Common Multiple (LCM) of two numbers, we need to identify the smallest positive integer that is divisible by both numbers in the pair. Let's go through the process step-by-step for each pair.
### Pair a: [tex]\( 108, 360 \)[/tex]
1. Prime Factorization:
- [tex]\( 108 = 2^2 \times 3^3 \)[/tex]
- [tex]\( 360 = 2^3 \times 3^2 \times 5 \)[/tex]
2. Identifying the highest powers of all prime factors present in either number:
- The highest power of [tex]\(2\)[/tex] is [tex]\(2^3\)[/tex] (as in 360).
- The highest power of [tex]\(3\)[/tex] is [tex]\(3^3\)[/tex] (as in 108).
- The highest power of [tex]\(5\)[/tex] is [tex]\(5\)[/tex] (since 5 is only present in 360).
3. Combine these highest powers to get the LCM:
[tex]\[
\text{LCM} = 2^3 \times 3^3 \times 5 = 8 \times 27 \times 5 = 1080
\][/tex]
So, the LCM of 108 and 360 is [tex]\( 1080 \)[/tex].
### Pair b: [tex]\( 34, 39 \)[/tex]
1. Prime Factorization:
- [tex]\( 34 = 2 \times 17 \)[/tex]
- [tex]\( 39 = 3 \times 13 \)[/tex]
2. Identifying the highest powers of all prime factors present in either number:
- The highest power of [tex]\(2\)[/tex] is [tex]\(2\)[/tex] (as in 34).
- The highest power of [tex]\(17\)[/tex] is [tex]\(17\)[/tex] (as in 34).
- The highest power of [tex]\(3\)[/tex] is [tex]\(3\)[/tex] (as in 39).
- The highest power of [tex]\(13\)[/tex] is [tex]\(13\)[/tex] (as in 39).
3. Combine these highest powers to get the LCM:
[tex]\[
\text{LCM} = 2 \times 17 \times 3 \times 13 = 34 \times 39 = 1326
\][/tex]
So, the LCM of 34 and 39 is [tex]\( 1326 \)[/tex].
### Conclusion
The LCMs of the given pairs are:
- For [tex]\( 108 \)[/tex] and [tex]\( 360 \)[/tex], the LCM is [tex]\( 1080 \)[/tex].
- For [tex]\( 34 \)[/tex] and [tex]\( 39 \)[/tex], the LCM is [tex]\( 1326 \)[/tex].
These solutions provide the smallest positive integers that are divisible by both numbers in each pair.
