Find the square root of the following number by the prime factorisation method. 5929
The Correct Answer and Explanation is:
To find the square root of 5929 using the prime factorization method, we need to express the number as a product of prime factors and then group the factors in pairs.
Step-by-Step Solution:
- Divide the number by the smallest prime number (2, 3, 5, 7, 11, etc.) until you can no longer divide. Start by checking divisibility by the smallest primes:
- 5929 is odd, so it is not divisible by 2.
- Sum of the digits of 5929: 5+9+2+9=255 + 9 + 2 + 9 = 255+9+2+9=25. Since 25 is not divisible by 3, 5929 is not divisible by 3.
- 5929 does not end in 0 or 5, so it is not divisible by 5.
- Check divisibility by 7. Divide 5929 by 7: 5929÷7=8475929 ÷ 7 = 8475929÷7=847 So, 5929 can be divided by 7. We now have: 5929=7×8475929 = 7 × 8475929=7×847
- Factor 847 further: Now, check if 847 is divisible by any prime numbers:
- 847 is odd, so not divisible by 2.
- Sum of the digits of 847: 8+4+7=198 + 4 + 7 = 198+4+7=19, which is not divisible by 3.
- 847 does not end in 0 or 5, so not divisible by 5.
- Check divisibility by 7: 847÷7=121847 ÷ 7 = 121847÷7=121 So, we now have: 847=7×121847 = 7 × 121847=7×121 Thus, we have: 5929=7×7×1215929 = 7 × 7 × 1215929=7×7×121
- Factor 121 further: 121 is a perfect square: 121=11×11121 = 11 × 11121=11×11
- Prime factorization of 5929:
Combining all the factors, we get: 5929=7×7×11×115929 = 7 × 7 × 11 × 115929=7×7×11×11 Or, in exponential form: 5929=72×1125929 = 7^2 × 11^25929=72×112 - Find the square root: The square root of a number is obtained by taking the square root of each factor: 5929=72×112=7×11=77\sqrt{5929} = \sqrt{7^2 × 11^2} = 7 × 11 = 775929=72×112=7×11=77
Conclusion:
The square root of 5929 is 77.
This method works because when you factorize a number and group the prime factors in pairs, you can take one factor from each pair to find the square root.
