Which numbers between 40 and 49 have 2 and 3 in their prime factorization?Use exponents to write the prime factorization of each number.
If it has 2 AND 3 in the prime factorization, it means that the number must be divisible by 2 AND by 3, which means that it is divisible by 6.
The number(s) between 50 and 59 that are divisable by 6 would be 54. The prime factorization of 54 is
54
=6*9
=2*3*3*3
=2133
You can attempt numbers between 40 and 49 in a similar way.
To find the numbers between 40 and 49 that have 2 and 3 in their prime factorization, we need to consider each number individually.
Start with the number 40. To determine its prime factorization, we divide it by the smallest prime number, which is 2, and continue dividing by 2 until we can no longer divide evenly.
40 ÷ 2 = 20
20 ÷ 2 = 10
10 ÷ 2 = 5
So, the prime factorization of 40 is 2^3 * 5. Now we check if 40 has the prime factor 3. Since 40 does not have 3 in its prime factorization, we move on to the next number.
Next, consider the number 41. To find its prime factorization, we follow the same process as before, starting with the smallest prime number, which is 2. However, because 41 is prime, it cannot be divided by 2 or any other prime number. So, the prime factorization of 41 is 41^1.
Moving on to the number 42. We divide it by 2 until we can no longer divide evenly.
42 ÷ 2 = 21
21 ÷ 3 = 7
The prime factorization of 42 is 2^1 * 3^1 * 7. Now we check if 42 has both 2 and 3 in its prime factorization, and indeed it does. So, 42 is a number between 40 and 49 that has both 2 and 3 in its prime factorization.
We can continue this process for the remaining numbers between 40 and 49, which are 43, 44, 45, 46, 47, 48, and 49. After finding the prime factorization of each number, we check if they have both 2 and 3 in their prime factorization.
Here's a summary of the results:
43: Prime number, so it doesn't have any other prime factors.
44: 2^2 * 11. It does not have 3 in its prime factorization.
45: 3^2 * 5. It has both 2 and 3 in its prime factorization.
46: 2 * 23. It does not have 3 in its prime factorization.
47: Prime number.
48: 2^4 * 3. It has both 2 and 3 in its prime factorization.
49: 7^2. It does not have 2 or 3 in its prime factorization.
So, the numbers between 40 and 49 that have 2 and 3 in their prime factorization are 42 and 48.