Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev

UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

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UPSC : Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev

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Square Root
The square root of a number is that number the product of which itself gives the given number, i.e, the square root of 400 is 20, the square root of 625 is 25.
The process of finding the square root is called evaluation. The square root of a number is denoted by the symbol called the radical sign(√ ).

How to Find the Square Root of an Integer?
(i) By the method of Prime Factors: When a given number is a perfect square, we resolve it into prime factors and take the product of prime factors, choosing one out of every two.
To find the square root of a perfect square by using the prime factorization method when a given number is a perfect square:
Step I: Resolve the given number into prime factors.
Step II: Make pairs of similar factors.
Step III: Take the product of prime factors, choosing one factor out of every pair.


Examples on square root of a perfect square by using the prime factorization method: 
1. Find the square root of 484 by prime factorization method.
Solution:
Resolving 484 as the product of primes, we get
Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev
484 = 2 × 2 × 11 × 11 
√484 = √(2 × 2 × 11 × 11
= 2 × 11 
Therefore, √484 = 22

2. Find the square root of 324.
Solution: The square root of 324 by prime factorization, we get 
Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev
324 = 2 × 2 × 3 × 3 × 3 × 3
√324 = √(2 × 2 × 3 × 3 × 3 × 3) 
= 2 × 3 × 3 
Therefore, √324 = 18 

(ii) By the method of Long Division: This method can be used when the number is large and the factors cannot be determined easily. This method can also be used when we want to add a least number or to subtract a least number from a given number so that the resulting number may give a perfect square of some number.
Steps of Long Division Method for Finding Square Roots:
Step I: Group the digits in pairs, starting with the digit in the units place. Each pair and the remaining digit (if any) is called a period.
Step II: Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor and also as the quotient.
Step III: Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.
Step IV: Now, the new divisor is obtained by taking two times the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.
Step V: Repeat steps (2), (3) and (4) till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.

Examples on square root of a perfect square by using the long division method
1. Find the square root of 784 by the long-division method.
Solution:
Marking periods and using the long-division method,
Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev
Therefore, √784 = 28

2. Evaluate √5329 using long-division method.
Solution: Marking periods and using the long-division method,
Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev
Therefore, √5329 =73

Properties of squares:
1. When a perfect square is written as a product of its prime factors each prime factor will appear an even number of times.

2. The difference between the squares of two consecutive natural numbers is always equal to the sum of the natural numbers. Thus, 412 – 402 = (40 + 41) = 81.
This property is very useful when used in the opposite direction—i.e. Given that the difference between the squares of two consecutive integers is 81, you should immediately realise that the numbers should be 40 and 41.

3. The square of a number ending in 1, 5 or 6 also ends in 1, 5 or 6 respectively.

4. The square of any number ending in 5: The last two digits will always be 25. The digits before that in the answer will be got by multiplying the digits leading up to the digit 5 in the number by 1 more than itself.

Illustration:
852 = ___25.
The missing digits in the above answer will be got by 8 × (8 + 1) = 8 × 9 = 72. Hence, the square of 85 is given by 7225.
Similarly, 1352 = ___25. The missing digits are 13 × 14 = 182. Hence, 1352 = 18225.

5. The value of a perfect square has to end in 1, 4, 5, 6, 9 or an even number of zeros.
In other words, a perfect square cannot end in 2, 3, 7, or 8 or an odd number of zeros.

6. If the units digit of the square of a number is 1, then the number should end in 1 or 9.

7. If the units digit of the square of a number is 4, then the units digit of the number is 2 or 8.

8. If the units digit of the square of a number is 9, then the units digit of the number is 3 or 7.

9. If the units of the square of a number is 6, then the unit’s digit of the number is 4 or 6.

10. The sum of the squares of the first ‘n’ natural numbers is given by
[(n) (n + 1) (2n + 1)]/6.

11. Normally, by squaring any number we increase the value of the number. The only integers for which this is not true are 0 and 1. (In these cases squaring the number has no effect on the value of the number).
Further, for values between 0 to 1, squaring the number reduces the value of the number. For example 0.52 < 0.5.

3. Square Root
If x2 = y, we say that the square root of y is x and we write y = x.

Thus, √4 = 2, √9 = 3, √196 = 14.

4. Cube Root:
Cube root of a is denoted as Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev
Thus, 8 = 2 x 2 x 2 = 2, 343 = 7 x 7 x 7 = 7 etc.

Important points to remember:
1. Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev

2. Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev
3. Number ending in 8 can never be a perfect square.
4. Remember the squares and cubes of 2 to 10. This will help in easily solving the problems.
Perfect Square - Number Theory, Quantitative Aptitude Quant Notes | EduRev

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