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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.
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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.
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.
How many prime numbers are on a die?
If you mean a regular six-sided die, there are 3: 2,3 and 5. Some people would argue that 1 is a prime number, but in fact 1 is not, as it only has one natural number divisor, while prime numbers have exactly two distinct natural number divisors.
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 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 Divisors In Math?
The divisors are the numbers in the outside of a division problem. example: 3)9 The three on the outside is the divisor The divisor is the adviser.
Which group sHow is all the whole number factors of a composite number 123458 16 or 1234561020 or 1234681224 or 123471428?
numbers 1, 2, 3, 4, 5, 8, 16 are divisors number 16 numbers 1, 2, 3, 4, 5, 6, 10, 20 are divisors number 20 numbers 1, 2, 3, 4, 6, 8, 12, 24 are divisors number 24 numbers 1, 2, 3, 4, 7, 12, 28 are divisors number 28 next number is 32 and divisors are 1, 2, 4, 8, 16, 32
What is the rule for 1 2 4 8 12 32?
Least number with exactly n even divisors 1 -> 0 divisor 2 -> 1 divisor 4 -> 2 divisors = 22 8 -> 3 divisors = 23 12 -> 4 divisors = 22x3 32 -> 5 divisors = 24
