Find the greatest perfect square that is a factor of the number
693
720
51,425
693 = 3^2 * 77
720 = 12^2 * 5
51425 = 55^2 * 17
None of these are a perfect square.
Hmm, let's see if I can square this puzzle. For 693, the factors are 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, and 693. And if we're searching for the greatest perfect square, we can rule out 1 as being too small, and 693 as being the actual number itself. After some clownish calculations, I'd say the greatest perfect square factor is 441, which is the square of 21.
To find the greatest perfect square that is a factor of a number, we need to prime factorize the given number and then identify the highest power of each prime factor that is a perfect square.
Let's go through each given number and find the greatest perfect square factor:
1. For the number 693:
- Start by determining the prime factorization of 693.
- Divide 693 by the smallest prime number, which is 3: 693 ÷ 3 = 231.
- Divide 231 by 3 again: 231 ÷ 3 = 77.
- Divide 77 by the smallest prime number, which is 7: 77 ÷ 7 = 11.
- Since 11 is a prime number, we can't divide it further.
- Now, let's consider the powers of the prime factors: 3^2 × 7^2 × 11^1.
- The highest power of each prime factor that is a perfect square is 3^2 (9) and 7^2 (49).
- Therefore, the greatest perfect square factor of 693 is 9 × 49 = 441.
2. For the number 720:
- Start by determining the prime factorization of 720.
- Divide 720 by the smallest prime number, which is 2: 720 ÷ 2 = 360.
- Divide 360 by 2 again: 360 ÷ 2 = 180.
- Divide 180 by 2 again: 180 ÷ 2 = 90.
- Divide 90 by the smallest prime number, which is 3: 90 ÷ 3 = 30.
- Divide 30 by the smallest prime number, which is 2: 30 ÷ 2 = 15.
- Divide 15 by the smallest prime number, which is 3: 15 ÷ 3 = 5.
- Since 5 is a prime number, we can't divide it further.
- Now, let's consider the powers of the prime factors: 2^4 × 3^2 × 5^1.
- The highest power of each prime factor that is a perfect square is 2^4 (16) and 3^2 (9).
- Therefore, the greatest perfect square factor of 720 is 16 × 9 = 144.
3. For the number 51,425:
- Start by determining the prime factorization of 51,425.
- Divide 51,425 by the smallest prime number, which is 5: 51,425 ÷ 5 = 10,285.
- Divide 10,285 by the smallest prime number, which is 5: 10,285 ÷ 5 = 2,057.
- Divide 2,057 by the smallest prime number, which is 7: 2,057 ÷ 7 = 293.
- Since 293 is a prime number, we can't divide it further.
- Now, let's consider the powers of the prime factors: 5^2 × 7^1 × 293^1.
- The highest power of each prime factor that is a perfect square is 5^2 (25).
- Therefore, the greatest perfect square factor of 51,425 is 25.
In conclusion:
- The greatest perfect square factor of 693 is 441.
- The greatest perfect square factor of 720 is 144.
- The greatest perfect square factor of 51,425 is 25.