Find some integers having 10 divisors.
what about 2x3x5x7x11x13x17x19....up to 10 prime numbers...that will get you more than 10
you can see from the first few, you also as a side, you get 6,10,14...15,21,33,...
If you want exactly 10 divisors, then the number will have to be a product of two primes, one raised to the 1st power, and the other raised to the 4th power.
For example,
567 = 3^4 * 7^1
Its factors are
1 3 7 9 21 27 63 81 189 567
see why at
http://www.wikihow.com/Find-How-Many-Factors-Are-in-a-Number
Sure, here are some integers that have exactly 10 divisors:
- The first one is 30, because its divisors are 1, 2, 3, 5, 6, 10, 15, and 30. That makes a total of 8 divisors, and if we include 1 and 30, we have a grand total of 10 divisors.
- The second one is 45, because its divisors are 1, 3, 5, 9, 15, and 45. If we double up these divisors and include 1 and 45, we again have a total of 10 divisors.
- Lastly, we have 120, which has divisors of 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. If we count these up, we have a grand total of 16 divisors. However, if we only include every other divisor, we’ll have 10 divisors.
Hope these integer suggestions tickle your mathematical funny bone!
To find integers with 10 divisors, we need to consider numbers that can be expressed as a product of two prime numbers raised to either the power of 9 or the power of 4 multiplied by another prime number.
Here are a few examples:
1. 2^9 * 3 = 5,184: This number has divisors 1, 2, 3, 6, 12, 16, 24, 48, 144, and 5184.
2. 2^4 * 5 = 80: This number has divisors 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80.
3. 2^4 * 7 = 112: This number has divisors 1, 2, 4, 7, 8, 14, 28, 56, 112.
4. 2^9 * 7 = 8,192: This number has divisors 1, 2, 4, 7, 8, 14, 16, 28, 32, and 8192.
5. 2^4 * 3^2 = 144: This number has divisors 1, 2, 3, 4, 6, 8, 9, 12, 16, and 144.
These are just a few examples, and there are infinitely many integers with 10 divisors.
To find integers that have exactly 10 divisors, we need to understand that the number of divisors a number has depends on its prime factorization.
First, let's consider the prime factorization of a number. Any positive integer can be expressed as a product of prime numbers raised to certain powers. For example, 24 can be written as 2^3 * 3^1 since 24 is divisible by 2 three times and by 3 once.
Now, let's focus on the number of divisors. If a number can be expressed as a product of primes raised to powers, the number of divisors it has can be obtained by taking the exponents of the prime factorization, increasing each exponent by 1, and then multiplying the results. For example, for 24 (2^3 * 3^1), the exponents are 3 and 1. Increasing these exponents by 1 gives 4 and 2. Multiplying 4 and 2 gives 8, so 24 has 8 divisors.
To find integers with exactly 10 divisors, we need to determine the prime factorization exponents that satisfy the equation (a + 1) * (b + 1) = 10, where a and b are the exponents.
Let's consider all possible combinations of the exponents that satisfy the equation:
1. (1 + 1) * (9 + 1) = 2 * 10 = 20
2. (2 + 1) * (4 + 1) = 3 * 5 = 15
3. (4 + 1) * (1 + 1) = 5 * 2 = 10
From these combinations, we can see that the possible exponents for the prime factors are (1, 9), (2, 4), and (4, 1).
Now, we just need to convert these exponents back into the actual numbers by raising the primes to those powers:
1. 2^1 * 3^9 = 19683
2. 2^2 * 3^4 = 144
3. 2^4 * 3^1 = 48
Therefore, the integers that have exactly 10 divisors are 19683, 144, and 48.