1)List all #s less than 100 with exactly 5 factors.
For this I got 16 , 36, 64, and 81.
2)Find the next 2 #s, each greater than 100, that have exactly 5 factors.
This is the problem I am having trouble with.
Are you including 1 and the number itself, or not? Either way, 36 has more than five factors: I can think of 2, 3, 4, 6, 9, 12 and 18. (Or do you mean prime factors?)
I am including 1 and the # itself, and you are right, 36 is not one of the answers. thank you for the correction, but i still need help with the second problem
See my response to your later repeat post.
To find the next two numbers, each greater than 100, that have exactly 5 factors, we need to understand the concept of factors.
Factors are the numbers that divide a given number evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because all these numbers divide 12 evenly.
To determine the number of factors a given number has, we factorize that number and count its prime factors. For example, to find the factors of 12, we factorize it as 2^2 * 3^1. The exponents of the prime factors are raised by 1 and multiplied together to give the total number of factors. In this case, it is (2+1) * (1+1) = 6.
Now, let's apply this concept to find the next two numbers with exactly 5 factors, each greater than 100:
1. Start with the number 101. We will check the factors for this number and determine if it has exactly 5 factors.
Factors of 101: 1, 101
As the number 101 only has two factors (1 and 101), it does not meet the requirement of exactly 5 factors.
2. Move on to the next number, which is 102. Let's factorize it and check the number of factors.
Factors of 102: 1, 2, 3, 6, 17, 34, 51, 102
The number 102 has 8 factors, so it does not meet the requirement of exactly 5 factors.
3. Continue this process by incrementing the numbers until we find two numbers that satisfy the requirement.
- For 103:
Factors of 103: 1, 103
Once again, the number 103 only has two factors, so it does not meet the requirement.
- For 104:
Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104
Similarly, the number 104 has 8 factors and does not meet the requirement.
4. Keep repeating this process until we find two numbers that have exactly 5 factors.
- For 105:
Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105
The number 105 has 8 factors, so it does not meet the requirement.
- For 106:
Factors of 106: 1, 2, 53, 106
Once again, the number 106 only has four factors, so it does not meet the requirement.
- For 107:
Factors of 107: 1, 107
The number 107 only has two factors and does not meet the requirement.
5. Continue this process until we find two numbers with exactly 5 factors.
- For 108:
Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
The number 108 has 12 factors and does not meet the requirement.
- For 109:
Factors of 109: 1, 109
The number 109 only has two factors and does not meet the requirement.
6. We can continue this process until we find two numbers with exactly 5 factors. However, this method can be time-consuming and may require checking multiple numbers.
To expedite the process, we can use the fact that numbers with exactly 5 factors are squares of prime numbers. In this case, we need to find two consecutive prime numbers after 10 (namely, 13 and 17) and square them:
- 13^2 = 169
- 17^2 = 289
Therefore, the next two numbers, each greater than 100, that have exactly 5 factors are 169 and 289.