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An integer (call it 'x') has exactly 3 divisors if and only if it is the square of a Prime number.
In other words, to generate a list of integers with exactly 3 divisors, just keep squaring prime numbers.
A number with 3 divisors cannot be prime (a prime number has only 2 divisors, 1 and itself). So it must be a composite number, which is a number that can be factored as a product of prime numbers (Fundamental Theorem of Arithmetic) -- i.e. a composite number must have at least one prime divisor.
In the case where the number has only 3 divisors, two of them are 1 and the number itself (neither of which are prime). Therefore the third divisor must be a prime number.
So the three divisors of 'x' are: 1, p, x where p is prime.
Now since p is a divisor (or factor) of x, and the only other divisor besides 1 and x itself, x must equal p*p -- or x=p^2 . Obvious x can't equal p*x and if x = p*1, x=p so x is prime, or has only 2 divisors...
If x = p^(3) , then x = p*p* p , or p*(p^2) ... this means that p^2 would also have to be a divisor of x, and this would contradict with x having only 3 divisors. For the same reason, x = p^(greater than 3) is also not possible.
So the only possibility is that an integer with exactly 3 divisors is the square of a prime number "p". The divisors are 1, p, and p^2.
I'm sure there's a simpler, more elegant way of explaining this, but it should be clear enough.
What else can I help you with?
What is a common divisor?
All numbers have divisors. Some numbers have some of the same divisors as other numbers. These are called common divisors. 3 is a divisor of 9. 3 is a divisor of 12. 3 is a common divisor of 9 and 12.
What is the smallest whole number that has exactly 6 distinct positive divisors?
The smallest whole number that has exactly 6 distinct positive divisors is 12. The divisors of 12 are 1, 2, 3, 4, 6, and 12. These are the only positive integers that evenly divide 12, making it the smallest number with exactly 6 distinct positive divisors.
Numbers with exactly 3 factors?
Squares of prime numbers have exactly three factors.
What is a positive divisor?
A divisor is another name for a factor. All counting numbers have divisors. In the equation 3 x 4 = 12, 3 and 4 are divisors of 12. -3 x -4 also equals 12, so some people distinguish between positive and negative divisors. A positive divisor is a divisor that is a positive number.
Which of the numbers 31 41 51 and 61 is a composite number?
A composite number is a positive integer greater than 1 that has more than two distinct positive divisors. In this case, the number 51 is a composite number because it has divisors other than 1 and itself, such as 3 and 17. Numbers 31, 41, and 61 are all prime numbers, as they only have two distinct positive divisors, 1 and the number itself.
What is a common divisor?
All numbers have divisors. Some numbers have some of the same divisors as other numbers. These are called common divisors. 3 is a divisor of 9. 3 is a divisor of 12. 3 is a common divisor of 9 and 12.
What is the third smallest integer with exactly 3 divisors?
25 is.
What numbers less than 18 have exactly four divisors such as the number 6 has these four divisors 1 2 3 and 6 List your answers as a comma-separated list?
6, 8, 10, 14, 15
Why 89 is a prime number?
The only numbers which divide into 89 exactly are 1 and 89. These are called factors or divisors. Any number which has exactly 2 divisors (1 and the number itself) is a prime number.
How do you use divisors?
Divisors are used to divide numbers.
What number has exactly 10 divisors?
The first number to have exactly 10 divisors is 48.
What are the divisors of 36 are also prime numbers?
2 and 3
What numbers have exactly 5 divisors?
16,81,12 well thats only i know
WHAT do Composite numbers have?
Composite numbers have more than two distinct positive divisors: 1 and the number itself. This means they can be divided evenly by at least one additional number. For example, the number 6 is composite because it can be divided by 1, 2, 3, and 6. In contrast, prime numbers have exactly two divisors.
What is the smallest number with exactly 14 divisors?
The smallest number with exactly 14 divisors is 192. The divisors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, and 192 itself.
What are the divisors of 9?
The divisors of 9 are the numbers that can divide 9 without leaving a remainder. In this case, the divisors of 9 are 1, 3, and 9. This is because 1 multiplied by 9 equals 9, and 3 multiplied by 3 equals 9. Therefore, the divisors of 9 are 1, 3, and 9.
What numbers can go into 18 and 96?
The numbers that can go into both 18 and 96 are their common divisors. The common divisors of 18 are 1, 2, 3, 6, 9, and 18, while the common divisors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. The common divisors of both numbers are 1, 2, 3, and 6.
