What is the largest number that can be divided into both 56 and 100?
To find the largest number that can be divided into both 56 and 100, we need to find the greatest common divisor (GCD) of these two numbers. Here's how:
Method 1: Prime Factorization
1. Find the prime factors of both numbers:
- Prime factorization of 56: 2 x 2 x 2 x 7 = 2^3 x 7
- Prime factorization of 100: 2 x 2 x 5 x 5 = 2^2 x 5^2
2. Take the common prime factors with the lowest exponent from the factorizations:
- Common prime factors: 2^2 = 4
3. Therefore, the largest number that can divide both 56 and 100 is 4.
Method 2: Euclidean Algorithm
1. Start with the two numbers: 56 and 100.
2. Divide the larger number (100) by the smaller number (56) and find the remainder:
- 100 ÷ 56 = 1 remainder 44
3. Now, divide the previous divisor (56) by the remainder (44) and find the new remainder:
- 56 ÷ 44 = 1 remainder 12
4. Repeat the process until the remainder is zero:
- 44 ÷ 12 = 3 remainder 8
- 12 ÷ 8 = 1 remainder 4
- 8 ÷ 4 = 2 remainder 0
5. At this point, the divisor immediately before the remainder becomes zero is the greatest common divisor (GCD).
- The GCD of 56 and 100 is 4.
Both methods give us the same result: The largest number that can divide both 56 and 100 is 4.