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The easy way to solve this question would be to use trial and error, but there is actually a simpler way to do this question. I started off with trial and error, because I expected the number to be very small, since squaring it can create large numbers and you can easily find factors after that, but I realized a pattern in my process of trial and error.
First of all, a prime number squared results in that number being the only prime. This is because the number itself is not divisible by anything. For example, if we had 7 squared, 49 does not have any factors that are prime except for 7. This works the same with other prime numbers. However, if the number is not a prime number, then it does not matter if the number is squared or not, so we can just take the number and see if it contains 3 different prime numbers or not.
The smallest 3 prime numbers are 2, 3, and 5, so this means that a number has to contain these 3 prime numbers, and that would be 30. Thus, 30 would be the number to square, and the number would then be 900. -Alan
The three smallest prime numbers are 2, 3 and 5. The LCM of these three numbers is 30. Using multiplication, I know that 30 is the square root of 900 (30*30). This means that 900 is a perfect square with 3 prime numbers as factors. I know that this is the smallest number of this sort because the square root of the perfect square has to be divisible by 3 prime numbers and the 30 is the first number that can, so 900 has to be the first perfect square that can be divided by 3 prime numbers.
900 is the smallest perfect square that can be divided by 3 prime numbers.
POTW: Info: - Smallest perfect square with 3 different prime factors?
Basically, what we are looking for here, is the smallest prime number that once prime factored, has the smallest three prime numbers (which would be 2, 3 and 5). If we multiply 2, 3 and 5, we get 30, which is the smallest number that has 3 different prime numbers, but it's not a perfect square. Instead of changing the prime numbers, we can square each number instead in a way that we get 2^2 x 3^2 x 5^2 which gives us 900 (a perfect square).
In order to solve this problem, I decided to start by using the first 3 prime factors (2,3,5) and multiplying them together, which gave me 30. However, because that isn't a perfect square, I decided to square that number again, as the resulting number would still have prime factors of 2,3 and 5, the smallest prime numbers. Because of this, the smallest perfect square with 3 prime numbers as its factors is 900.
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ReplyDeleteThe easy way to solve this question would be to use trial and error, but there is actually a simpler way to do this question. I started off with trial and error, because I expected the number to be very small, since squaring it can create large numbers and you can easily find factors after that, but I realized a pattern in my process of trial and error.
ReplyDeleteFirst of all, a prime number squared results in that number being the only prime. This is because the number itself is not divisible by anything. For example, if we had 7 squared, 49 does not have any factors that are prime except for 7. This works the same with other prime numbers. However, if the number is not a prime number, then it does not matter if the number is squared or not, so we can just take the number and see if it contains 3 different prime numbers or not.
The smallest 3 prime numbers are 2, 3, and 5, so this means that a number has to contain these 3 prime numbers, and that would be 30. Thus, 30 would be the number to square, and the number would then be 900.
-Alan
The three smallest prime numbers are 2, 3 and 5. The LCM of these three numbers is 30. Using multiplication, I know that 30 is the square root of 900 (30*30). This means that 900 is a perfect square with 3 prime numbers as factors. I know that this is the smallest number of this sort because the square root of the perfect square has to be divisible by 3 prime numbers and the 30 is the first number that can, so 900 has to be the first perfect square that can be divided by 3 prime numbers.
ReplyDelete900 is the smallest perfect square that can be divided by 3 prime numbers.
POTW:
ReplyDeleteInfo:
- Smallest perfect square with 3 different prime factors?
Basically, what we are looking for here, is the smallest prime number that once prime factored, has the smallest three prime numbers (which would be 2, 3 and 5). If we multiply 2, 3 and 5, we get 30, which is the smallest number that has 3 different prime numbers, but it's not a perfect square. Instead of changing the prime numbers, we can square each number instead in a way that we get 2^2 x 3^2 x 5^2 which gives us 900 (a perfect square).
Grade 7/8 POTW
ReplyDeleteIn order to solve this problem, I decided to start by using the first 3 prime factors (2,3,5) and multiplying them together, which gave me 30. However, because that isn't a perfect square, I decided to square that number again, as the resulting number would still have prime factors of 2,3 and 5, the smallest prime numbers. Because of this, the smallest perfect square with 3 prime numbers as its factors is 900.
The smallest perfect square with 3 prime factors is 900
ReplyDeleteThe smallest perfect square that has 3 prime numbers as factors is 900. Those 3 prime numbers are 2, 3, and 5.
ReplyDelete