Sure, let's solve these problems step by step.
### Finding the Largest Number that Divides 48 and 84
To find the largest number that divides both 48 and 84 exactly, we need to determine their Greatest Common Divisor (GCD).
1. Prime Factorization:
- Prime factorization of 48: [tex]\( 48 = 2^4 \times 3 \)[/tex]
- Prime factorization of 84: [tex]\( 84 = 2^2 \times 3 \times 7 \)[/tex]
2. Common Factors:
- The common prime factors between 48 and 84 are [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
- The lowest powers of these common prime factors are:
- [tex]\(2^2\)[/tex] from [tex]\(84\)[/tex] (since [tex]\(84\)[/tex] has only [tex]\(2^2\)[/tex], and [tex]\(2^2 \leq 2^4\)[/tex])
- [tex]\(3\)[/tex] from both (since both have [tex]\(3\)[/tex] to the power of 1)
3. Compute the GCD:
[tex]\[
\text{GCD} = 2^2 \times 3 = 4 \times 3 = 12
\][/tex]
So, the largest number that divides 48 and 84 exactly is 12.
### Finding the Greatest Number of People to Share 25 Oranges and 30 Peoplen Equally
To determine the greatest number of people that can share 25 oranges and 30 people equally, we need to find the GCD of 25 and 30.
1. Prime Factorization:
- Prime factorization of 25: [tex]\( 25 = 5^2 \)[/tex]
- Prime factorization of 30: [tex]\( 30 = 2 \times 3 \times 5 \)[/tex]
2. Common Factors:
- The common prime factor between 25 and 30 is [tex]\(5\)[/tex].
- The lowest power of this common prime factor is [tex]\(5^1\)[/tex] (since [tex]\(5\)[/tex] is to the power of 1 in [tex]\(30\)[/tex] and [tex]\(1 \leq 2\)[/tex])
3. Compute the GCD:
[tex]\[
\text{GCD} = 5^1 = 5
\][/tex]
So, the greatest number of people that can share 25 oranges and 30 exactly is 5.
### Summary:
- The largest number that divides 48 and 84 exactly is 12.
- The greatest number of people that can share 25 oranges and 30 exactly is 5.
