find the gcf and lcm of 315 and 882.
factor both of them
315 = 5*63 = 5*7*9
= 3*3*5*7
882 = 2*441 = 2*7*63 = 2*7*7*9
= 2*3*3*7*7
now you can see the answers
they each have
2 threes and 1 seven = 63
315 needs 2 and 7 so 315*2*7 = 4410
882 needs 5 so 4410 sure enough
To find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of 315 and 882, we can use a combination of factorization and multiplication.
Step 1: Prime Factorization
Let's begin by finding the prime factors of both numbers.
For 315:
Start by dividing it by the smallest prime number, which is 2. Since 315 is an odd number, we move to the next prime number, which is 3.
315 ÷ 3 = 105
Next, divide 105 by 3 again:
105 ÷ 3 = 35
The only prime factor left is 5, so we divide 35 by 5:
35 ÷ 5 = 7
So, the prime factorization of 315 is 3 × 3 × 5 × 7, which can be written as 3^2 × 5 × 7.
For 882:
Divide it by 2 repeatedly until we cannot divide any further:
882 ÷ 2 = 441
441 ÷ 3 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
So, the prime factorization of 882 is 2 × 3^2 × 7^2.
Step 2: GCF
To find the GCF, we take the product of the common prime factors with the smallest exponent.
Common prime factors are: 3 × 7
The exponents are: 1 for 3 (from 3^1) and 1 for 7 (from 7^1).
So, the GCF of 315 and 882 is 3 × 7 = 21.
Step 3: LCM
To find the LCM, we take the product of all prime factors with the greatest exponent from either number.
All prime factors are: 2, 3, 5, and 7.
The greatest exponent for each prime factor is: 1 for 2, 2 for 3 (from 3^2), 1 for 5, and 2 for 7 (from 7^2).
So, the LCM of 315 and 882 is 2 × 3^2 × 5 × 7^2 = 17640.
Therefore, the GCF is 21, and the LCM is 17640.