use the prime factorization of each pair of numbers to identify three common factors and three common multiples.
a) 14 and 84
(please show me how to get the 3 common multiples)
thank you!!!
well,both are even, try 2
7,42
try 7
1,6
what about 2x7, or 14?
36
To find the prime factorization of a number, you need to factorize it into its prime factors. Here's how you can find the prime factorization of the numbers 14 and 84:
Prime factorization of 14:
Start by dividing 14 by the smallest prime number, which is 2. Since 14 is divisible by 2, we get 14 ÷ 2 = 7. Now, 7 is a prime number, so the prime factorization of 14 is 2 × 7.
Prime factorization of 84:
Similarly, start by dividing 84 by 2. You get 84 ÷ 2 = 42. Now, continue dividing 42 by 2. You get 42 ÷ 2 = 21. Next, divide 21 by 3, which gives you 21 ÷ 3 = 7. Finally, 7 is a prime number, so the prime factorization of 84 is 2 × 2 × 3 × 7.
Now that we have the prime factorization of both numbers, we can identify the common factors and multiples.
Three common factors:
To find the common factors, we look for prime factors that are shared by both numbers. In this case, the only prime factor that is common to both 14 and 84 is 7. So, the three common factors are 1, 7, and 7.
Three common multiples:
To find the common multiples, we need to determine the least common multiple (LCM) of the two numbers. The LCM is the smallest multiple that both numbers share. To calculate the LCM, we take the highest power of each prime factor that appears in their prime factorizations.
Prime factorization of 14: 2 × 7
Prime factorization of 84: 2 × 2 × 3 × 7
Now, take the highest power of each prime factor that appears in the factorizations:
Highest power of 2: 2^2 = 4
Highest power of 3: 3^1 = 3
Highest power of 7: 7^1 = 7
To find the LCM, multiply the highest powers together: 4 × 3 × 7 = 84.
So, the three common multiples of 14 and 84 are 84, 168, and 252.