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Factors
Perfect squares have an odd number of factors. If you think of factor pairs, the pair of factor that when multiplied together equals the given number, you realize that there are two factors for each pair until you come to the factor that when squared equals the number. For example, the factor pairs of 16 are 1 x 16, 2 x 8, and 4 x 4, giving us the factors of 1, 2, 4, 8, and 16 - an odd number of factors.
Prime Factors
However, the situation is different with prime factors. It will depend on whether you list every prime factor or only the distinct prime factors. There will always be an even number of prime factors because when you square a number, you double the prime factors, which means you have an even number. That is not the case for distinct prime factors, because when you square a number, you are not adding any additional distinct factors. So, in some cases, there will be an even number of distinct prime factors and sometimes an odd number. Some examples follow.
4 = 2 x 2. There are two prime factors, but one distinct prime factor.
36 = 2 x 2 x 3 x 3. There are four prime factors and two distinct prime factors.
900 = 2 x 2 x 3 x 3 x 5 x 5. There are six prime factors and three distinct prime factors.
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Do perfect squares alternate between odd and even?
yes they do alternate
How many positive integers less than 2008 have an even number of divisors?
There are 1,963 such integers. Every factor of a number has a pair. The only time there will be an odd number of factors is if one factor is repeated, ie the number is a perfect square. So the question is really asking: how many positive integers less than 2008 (in the range 1 to 2007) are not perfect squares. √2007 = 44 and a bit (it lies between 44 and 45) So there are 44 integers less than (or equal to) 2007 which are perfect squares → 2007 - 44 = 1963 integers are not perfect squares in the range 1-2007 and have an even number of factors (divisors).
Number of factors of 400 that are perfect squares?
400 can be factored as 24 * 52, or 22 * 22 * 52. A perfect square factor of 400 will always have an even exponent, so here is a list: 12, 22, 42, 52, 102, and 202.
What is the smallest number that divides 61440 to give a perfect square?
The prime factors of 61440 are, 212 x 3 x 5. As a square number must possess prime factors with even numbered exponents then division by 15 will leave 212 which is (26)². Thus, 61440 ÷ 15 = 4096 = 64²
What numbers have an even amount of factors?
If it's just factors, then any number have an even amount of factors. Let a be any number, a = 1 a so a have two factors, 1 and a. Two is even. But if the question is what number have an even amount of PRIME factors, then it is different. Because if a is not prime, the might exist another two numbers b and c != 1 such that a = b c. and also, b is not prime, b = d e So a = bc = de c = c d e So a have three prime factors (1 is not prime for some reason) Then this question is less trivial: 6 have only prime factor 2 and 3, 10 only have prime factor 2 and 5, they are even. but 9 only have prime factor 3, so it's not an even amount, so 9 is not.
What numbers always have an even number of factors?
All positive integers which are not perfect squares.
What are all numbers with an even amount of factors?
All compound numbers that are not perfect squares.
What even number less than 100 has an odd number of factors?
Perfect squares have odd numbers of factors. The perfect squares less than 100 are: 1,4,9,16,25,36,49,64,81,100. 64 seems to fit both criteria.
Why do perfect squares have an odd number of factors?
Because the square root isn't listed twice.
How can I find the perfect square factors of a large number such as 193440000?
You can find perfect square factors from the prime factorization. The prime factorization of 193440000 is 28 x 3 x 54 x 13 x 31. The even powers are squares. 22, 24, 26, 28, 52, 54, (2 x 5)2, (2 x 5)4 Don't forget 1. 1, 4, 16, 25, 64, 100, 256, 625, 10000 are the perfect square factors of 193440000
Are perfect squares always even?
Yes they are always even, other wise it would not be a perfect sqare.
Why is the square root of 36 not an irrational number?
Because it is an integer, 6. The underlying reason is that all of its prime factors are expressed as squares. 36 = 22 x 32If all prime factors do not have an even exponent, the square root will not be an integer.
Do perfect squares alternate between odd and even?
yes they do alternate
Numbers that have an odd number of factors are?
Perfect squares ( also called square numbers) have an odd number of factors and primes squared have 3 factors. Brief Explanation: If you start with a prime number, it has 2 factors by definition. Square that number and you have 3 factors, which is an odd number. So primes squared always have an odd number of factors. For example, 5 has 1 and 5 as factors, 25 has 1,5, and 25. What about an odd number such as 21 which is not the square of a prime. It has factors 1, 21, 3 and 7 so an even number of factors. How about 27, 1,27, 3, 9 once again even. What I was trying to show is that factors of numbers come in pairs and so only certain numbers will have an odd number of factors. Let's look at one more perfect square that is not a prime squared. How about 16 which is 4 squared. The factors are 1,2,4,8,and 16 which is an odd number of factors. Looking at these as pairs we see the factor pairs of 16 are 1 x 16, 2 x 8, and 4 x 4, giving us the factors of 1, 2, 4, 8, and 16 - an odd number of factors. So we conclude that perfect squares have an odd number of factors and primes squared have 3 factors.
How many positive integers less than 2008 have an even number of divisors?
There are 1,963 such integers. Every factor of a number has a pair. The only time there will be an odd number of factors is if one factor is repeated, ie the number is a perfect square. So the question is really asking: how many positive integers less than 2008 (in the range 1 to 2007) are not perfect squares. √2007 = 44 and a bit (it lies between 44 and 45) So there are 44 integers less than (or equal to) 2007 which are perfect squares → 2007 - 44 = 1963 integers are not perfect squares in the range 1-2007 and have an even number of factors (divisors).
What multiples of 8 are perfect squares?
Any even exponent of 8.
What do you notice about the number of factors each of 12 14 20 4 9 and 16?
12, 14, and 20 have an even number of factors. 4, 9, and 16 also have an even number of factors, but since they're perfect squares, two of the factors are the same number in each case, so each appears to have an odd number of factors.
