To determine the largest number that has the multiples 168 and 288, we need to find the greatest common divisor (GCD) of these two numbers. The GCD is the largest number that can divide both 168 and 288 without leaving a remainder. Here’s a step-by-step solution to find this GCD:
1. Prime Factorization:
First, we perform the prime factorization of each number.
- For 168:
- 168 is even, so divide by 2: [tex]\( 168 \div 2 = 84 \)[/tex]
- 84 is even, so divide by 2: [tex]\( 84 \div 2 = 42 \)[/tex]
- 42 is even, so divide by 2: [tex]\( 42 \div 2 = 21 \)[/tex]
- 21 is divisible by 3 (the next smallest prime): [tex]\( 21 \div 3 = 7 \)[/tex]
- 7 is a prime number.
Thus, the prime factors of 168 are [tex]\( 2^3 \times 3 \times 7 \)[/tex].
- For 288:
- 288 is even, so divide by 2: [tex]\( 288 \div 2 = 144 \)[/tex]
- 144 is even, so divide by 2: [tex]\( 144 \div 2 = 72 \)[/tex]
- 72 is even, so divide by 2: [tex]\( 72 \div 2 = 36 \)[/tex]
- 36 is even, so divide by 2: [tex]\( 36 \div 2 = 18 \)[/tex]
- 18 is even, so divide by 2: [tex]\( 18 \div 2 = 9 \)[/tex]
- 9 is divisible by 3: [tex]\( 9 \div 3 = 3 \)[/tex]
- 3 is divisible by 3: [tex]\( 3 \div 3 = 1 \)[/tex]
Thus, the prime factors of 288 are [tex]\( 2^5 \times 3^2 \)[/tex].
2. Finding the Common Prime Factors:
Identify the prime factors common to both factorizations and take the lowest power of each.
- The common prime factors between 168 ([tex]\(2^3 \times 3 \times 7\)[/tex]) and 288 ([tex]\(2^5 \times 3^2\)[/tex]) are [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
- The lowest power of 2 common to both is [tex]\(2^3\)[/tex].
- The lowest power of 3 common to both is [tex]\(3^1\)[/tex].
3. Calculating the GCD:
Multiply the common prime factors together to get the GCD.
- GCD = [tex]\(2^3 \times 3^1\)[/tex]
- [tex]\(2^3 = 8\)[/tex]
- [tex]\(3^1 = 3\)[/tex]
- [tex]\(8 \times 3 = 24\)[/tex]
Hence, the greatest common divisor of 168 and 288 is 24. This means the largest number that divides both 168 and 288 without a remainder is 24.
