3. Find the GCF and LCM for the following set of numbers.
100 126 250
To find the GCF (Greatest Common Factor) and LCM (Least Common Multiple) of a set of numbers, we need to find the factors of each number.
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Factors of 250: 1, 2, 5, 10, 25, 50, 125, 250
Now, let's find the GCF by finding the common factors among the numbers:
The common factors of 100, 126, and 250 are 1 and 2.
Therefore, the GCF of 100, 126, and 250 is 2.
To find the LCM, we need to find the smallest multiple that is divisible by all three numbers:
Multiples of 100: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, ...
Multiples of 126: 126, 252, 378, 504, 630, 756, 882, 1008, ...
Multiples of 250: 250, 500, 750, 1000, ...
From the multiples listed, we can see that the smallest multiple that is divisible by all three numbers (100, 126, and 250) is 1000.
Therefore, the LCM of 100, 126, and 250 is 1000.
To find the GCF (Greatest Common Factor) and LCM (Least Common Multiple) for the set of numbers {100, 126, 250}, we will follow the steps below:
Step 1: Prime Factorization
- Prime factorize each number in the set.
Prime factorization of 100: 2^2 * 5^2
Prime factorization of 126: 2 * 3^2 * 7
Prime factorization of 250: 2 * 5^3
Step 2: GCF
- The GCF is the product of the common prime factors raised to their lowest exponent.
The common prime factors in this case are 2 and 5.
- The lowest exponent for 2 is 1 (since it appears in all three numbers).
- The lowest exponent for 5 is 1 (since it appears in two numbers, but not in all three).
So, GCF = 2^1 * 5^1 = 10.
Step 3: LCM
- The LCM is the product of all prime factors raised to their highest exponent.
In this case, the prime factors are 2, 3, 5, and 7.
- 2 has the highest exponent of 2.
- 3 has the highest exponent of 2 (since it appears in one number with an exponent of 2).
- 5 has the highest exponent of 3 (since it appears in one number with an exponent of 3).
- 7 has the highest exponent of 1.
So, LCM = 2^2 * 3^2 * 5^3 * 7 = 12600.
Therefore, the GCF for the given set of numbers {100, 126, 250} is 10, and the LCM is 12600.
To find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of a set of numbers, you need to first prime factorize each number.
1. Prime factorize 100:
100 can be written as 2^2 * 5^2
2. Prime factorize 126:
126 can be written as 2 * 3^2 * 7
3. Prime factorize 250:
250 can be written as 2 * 5^3
To find the GCF, you need to identify the common prime factors and their smallest exponent. The GCF is the product of these common prime factors raised to their smallest exponent.
4. Identify the common prime factors:
The common prime factors are 2 and 5.
5. Identify the smallest exponents of the common prime factors:
For 2, the smallest exponent is 1.
For 5, the smallest exponent is 2.
6. Calculate the GCF:
GCF = 2^1 * 5^2 = 10
To find the LCM, you need to identify the maximum exponent of each prime factor present in any of the numbers. The LCM is the product of these prime factors raised to their maximum exponent.
7. Identify the maximum exponents of the prime factors:
For 2, the maximum exponent is 2.
For 3, the maximum exponent is 2.
For 5, the maximum exponent is 3.
For 7, the maximum exponent is 1.
8. Calculate the LCM:
LCM = 2^2 * 3^2 * 5^3 * 7^1 = 6300
So, the GCF of 100, 126, and 250 is 10, and the LCM is 6300.