What is the greatest perfect square that is a factor of 1290
To find the greatest perfect square that is a factor of 1290, we need to factorize 1290.
First, let's find the prime factorization of 1290: 1290 = 2 * 3 * 5 * 43.
Now, we need to determine the largest square that can be formed using the prime factors.
We can form a square using the factors 2 * 3 * 5 = 30, which is a perfect square because each prime factor has an even exponent. However, we need to check if the factor 43 is squared as well.
Since 43 * 43 = 1849, which is greater than 1290, we can't form a square using the factor 43.
Thus, the greatest perfect square that is a factor of 1290 is 30.
To find the greatest perfect square that is a factor of 1290, we can first find the prime factorization of 1290.
The prime factorization of 1290 is:
1290 = 2 * 3 * 5 * 43
Now, let's group the factors into pairs of the same prime factors:
1290 = (2 * 3) * (5) * (43)
Since we want a perfect square, we need pairs of prime factors. In this case, we have a pair of 2s and a pair of 3s. We can ignore the factor 5 and 43 as they are not paired.
Now, let's square the pairs:
Pair of 2s: (2 * 2) = 4
Pair of 3s: (3 * 3) = 9
The greatest perfect square that is a factor of 1290 is 4 * 9 = 36.
To find the greatest perfect square that is a factor of 1290, we need to factorize 1290 and identify the perfect squares among its factors.
Step 1: Prime factorize 1290
We can start by dividing 1290 by its smallest prime factor, which is 2:
1290 ÷ 2 = 645
Now, let's continue factoring 645:
645 ÷ 3 = 215
215 is not divisible by 2 or 3, so we move on to the next prime factor, which is 5:
215 ÷ 5 = 43
Now, we have found the prime factorization of 1290:
1290 = 2 × 3 × 5 × 43
Step 2: Identify the perfect squares among the factors
Among the prime factors, 2 and 5 are not perfect squares. However, 3 and 43 are prime numbers and could potentially be perfect squares.
A number is a perfect square if all its prime factors occur in pairs of even exponents. Since 3 is a prime factor of 1290 with an odd exponent, it is not a perfect square. On the other hand, 43 is a prime factor with an exponent of 1, which is an odd exponent. So, 43 is also not a perfect square.
Step 3: Find the product of the perfect square factors
Since there are no perfect squares among the factors of 1290, the greatest perfect square that is a factor of 1290 is 1.
In conclusion, the greatest perfect square that is a factor of 1290 is 1.