To find the greatest number that divides both 198 and 360 without leaving a remainder, we need to determine their greatest common divisor (GCD). Here's the step-by-step process for finding the GCD:
1. Prime Factorization:
- First, we need to find the prime factorization of each number.
Prime Factorization of 198:
- 198 is even, so we start by dividing by 2: [tex]\(198 ÷ 2 = 99\)[/tex]
- 99 is divisible by 3: [tex]\(99 ÷ 3 = 33\)[/tex]
- 33 is also divisible by 3: [tex]\(33 ÷ 3 = 11\)[/tex]
- 11 is a prime number.
Therefore, the prime factors of 198 are: [tex]\(2, 3^2, 11\)[/tex]
Prime Factorization of 360:
- 360 is even, so we start by dividing by 2: [tex]\(360 ÷ 2 = 180\)[/tex]
- 180 is even, divide by 2: [tex]\(180 ÷ 2 = 90\)[/tex]
- 90 is even, divide by 2: [tex]\(90 ÷ 2 = 45\)[/tex]
- 45 is divisible by 3: [tex]\(45 ÷ 3 = 15\)[/tex]
- 15 is also divisible by 3: [tex]\(15 ÷ 3 = 5\)[/tex]
- 5 is a prime number.
Therefore, the prime factors of 360 are: [tex]\(2^3, 3^2, 5\)[/tex]
2. Identify Common Prime Factors and their Lowest Exponents:
- The common prime factors between 198 and 360 are [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
- For [tex]\(2\)[/tex], in 198 it appears with exponent [tex]\(1\)[/tex], and in 360 it appears with exponent [tex]\(3\)[/tex]. Therefore, the lowest exponent is [tex]\(1\)[/tex].
- For [tex]\(3\)[/tex], it appears in both numbers with exponent [tex]\(2\)[/tex].
3. Calculate the GCD:
- Multiply the common prime factors with their lowest exponents:
[tex]\[
\text{GCD} = 2^1 \times 3^2 = 2 \times 9 = 18
\][/tex]
Thus, the greatest number that divides both 198 and 360 without leaving a remainder is [tex]\( \boxed{18} \)[/tex].
