Finding The Least Number To Add Or Subtract For Perfect Squares

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In the realm of mathematics, perfect squares hold a special place. A perfect square is an integer that can be expressed as the square of another integer. For instance, 9 is a perfect square because it is the result of 3 squared (3 * 3 = 9). Understanding perfect squares is crucial in various mathematical applications, including algebra, geometry, and number theory. This article delves into the process of finding the least number that must be subtracted from or added to a given number to transform it into a perfect square. We will explore various examples and step-by-step solutions to elucidate this concept.

The central theme of this discussion revolves around two primary types of problems:

  1. Finding the least number that must be subtracted from a given number to make it a perfect square.
  2. Finding the least number that must be added to a given number to make it a perfect square.

These problems require a methodical approach, often involving the long division method to find the square root of the given number. By analyzing the remainder or the difference from the next perfect square, we can determine the required number to be subtracted or added. The following sections will elaborate on the methods and techniques used to solve these problems effectively. Understanding these methods is essential for anyone looking to enhance their problem-solving skills in number theory and arithmetic.

1. Finding the Least Number to Subtract

The first type of problem we address is finding the smallest number to subtract from a given number to obtain a perfect square. This involves a process of trial and error combined with the understanding of square roots. The most efficient method to tackle this is by using the long division method to calculate the square root of the given number. The remainder obtained in this process is the number that needs to be subtracted to make the original number a perfect square. To further illustrate, we will explore detailed examples with step-by-step explanations to ensure a thorough understanding of the concept.

Example Problems: Subtracting to Achieve Perfect Squares

Let's delve into some examples to understand how to find the least number to subtract from a given number to make it a perfect square. We will apply the long division method for finding square roots and then interpret the remainder to solve the problem. These examples will provide a clear understanding of the practical application of this mathematical concept. Each example is broken down step-by-step to facilitate easier comprehension and problem-solving skills.

(i) 1,989

To find the least number to subtract from 1,989 to make it a perfect square, we follow these steps:

  1. Use the long division method to find the square root of 1,989.
  2. We pair the digits from the right: 19 89.
  3. Find the largest number whose square is less than or equal to 19. That number is 4 (4² = 16).
  4. Divide 19 by 4, giving a quotient of 4 and a remainder of 3. Bring down the next pair (89) to make the new dividend 389.
  5. Double the quotient (4) to get 8. Find a digit X such that 8X * X is less than or equal to 389. Through trial, we find that X = 4 works (84 * 4 = 336).
  6. Subtract 336 from 389, which gives a remainder of 53.

The remainder, 53, is the number that needs to be subtracted from 1,989 to make it a perfect square.

Therefore, 1,989 - 53 = 1,936, and the square root of 1,936 is 44.

(ii) 1,19,766

Now, let's find the least number to subtract from 1,19,766 to make it a perfect square. We employ the same long division method:

  1. Apply the long division method to find the square root of 1,19,766.
  2. Pair the digits from the right: 1 19 76 6.
  3. The largest number whose square is less than or equal to 1 is 1 (1² = 1).
  4. Bring down the next pair (19). The new dividend is 19.
  5. Double the quotient (1) to get 2. Find a digit X such that 2X * X is less than or equal to 19. We find X = 4 (24 * 4 = 96).
  6. Subtract 96 from 119, giving a remainder of 23. Bring down the next pair (76) to make the new dividend 2376.
  7. Double the current quotient (14) to get 28. Find a digit X such that 28X * X is less than or equal to 2376. Through trial, we find X = 8 (288 * 8 = 2304).
  8. Subtract 2304 from 2376, giving a remainder of 72. Bring down the last digit (6) to make the new dividend 726.
  9. Double the current quotient (148) to get 296. Find a digit X such that 296X * X is less than or equal to 726. Through trial, we find X = 2 (2962 * 2 = 5924). This is too big so we try 296 *2 = 592
  10. Subtract 592 from 726 giving a remainder of 134.

The remainder, 134, is the number that needs to be subtracted from 1,19,766 to make it a perfect square.

Thus, 1,19,766 - 134 = 1,19,632, which is the perfect square of 346.

(iii) 6,249

To determine the least number to subtract from 6,249 to make it a perfect square:

  1. Employ the long division method for finding the square root of 6,249.
  2. Pair the digits from the right: 62 49.
  3. The largest number whose square is less than or equal to 62 is 7 (7² = 49).
  4. Subtract 49 from 62, giving a remainder of 13. Bring down the pair 49 to make the new dividend 1349.
  5. Double the quotient (7) to get 14. Find a digit X such that 14X * X is less than or equal to 1349. We find X = 9 (149 * 9 = 1341).
  6. Subtract 1341 from 1349, giving a remainder of 8.

The remainder, 8, is the number that needs to be subtracted from 6,249 to make it a perfect square.

Hence, 6,249 - 8 = 6,241, which is the perfect square of 79.

(iv) 1,525

To find the least number to subtract from 1,525 to make it a perfect square:

  1. Utilize the long division method to find the square root of 1,525.
  2. Pair the digits from the right: 15 25.
  3. The largest number whose square is less than or equal to 15 is 3 (3² = 9).
  4. Subtract 9 from 15, giving a remainder of 6. Bring down the pair 25 to make the new dividend 625.
  5. Double the quotient (3) to get 6. Find a digit X such that 6X * X is less than or equal to 625. We find X = 9 (69 * 9 = 621).
  6. Subtract 621 from 625, giving a remainder of 4.

The remainder, 4, is the number that needs to be subtracted from 1,525 to make it a perfect square.

Therefore, 1,525 - 4 = 1,521, which is the perfect square of 39.

(v) 2,73,682

To find the least number to subtract from 2,73,682 to achieve a perfect square:

  1. Apply the long division method to find the square root of 2,73,682.
  2. Pair the digits from the right: 2 73 68 2.
  3. The largest number whose square is less than or equal to 2 is 1 (1² = 1).
  4. Subtract 1 from 2, giving a remainder of 1. Bring down the pair 73 to make the new dividend 173.
  5. Double the quotient (1) to get 2. Find a digit X such that 2X * X is less than or equal to 173. We find X = 6 (26 * 6 = 156).
  6. Subtract 156 from 173, giving a remainder of 17. Bring down the pair 68 to make the new dividend 1768.
  7. Double the current quotient (16) to get 32. Find a digit X such that 32X * X is less than or equal to 1768. We find X = 5 (325 * 5 = 1625).
  8. Subtract 1625 from 1768, giving a remainder of 143. Bring down the last digit (2) to make the new dividend 1432.
  9. Double the current quotient (165) to get 330. Find a digit X such that 330X * X is less than or equal to 1432. We find X = 4 (3304 * 4 = 13216). This is too big, so we try X=3, (3303 * 3 = 9909). Again too big, so we try 330 * 0 = 0.
  10. Try 3304 * 4 = 13216 which is too big. So the next smaller number is 3304 * 0 = 0
  11. Subtract 0 from 1432 giving a remainder of 1432.

The remainder, 1432, is the number that needs to be subtracted from 2,73,682 to make it a perfect square.

Thus, 2,73,682 - 1432 = 2,72,250, which is the perfect square of 522.

2. Finding the Least Number to Add

Now, let's shift our focus to finding the least number that must be added to a given number to make it a perfect square. This task also involves the long division method, but with a slightly different approach. After finding the quotient through long division, we need to determine the next higher integer and square it. The difference between this squared value and the original number will give us the number to be added. We will illustrate this method through examples, providing step-by-step explanations for clarity and understanding.

Example Problems: Adding to Achieve Perfect Squares

To better understand this concept, let's solve a few example problems. These examples will demonstrate the process of finding the least number to be added to a given number to make it a perfect square. The steps involve using the long division method to find an approximate square root, then calculating the difference needed to reach the next perfect square. Each example will be broken down methodically for ease of understanding.

(i) 8210

To find the least number to be added to 8210 to make it a perfect square, we proceed as follows:

  1. Use the long division method to find the square root of 8210.
  2. Pair the digits from the right: 82 10.
  3. The largest number whose square is less than or equal to 82 is 9 (9² = 81).
  4. Subtract 81 from 82, giving a remainder of 1. Bring down the pair 10 to make the new dividend 110.
  5. Double the quotient (9) to get 18. Find a digit X such that 18X * X is less than or equal to 110. We find X = 0 (180 * 0 = 0 so 0 works better than 181 * 1 = 181 which is too big. So quotient is 90).
  6. Thus the quotient is 90, and since there is a non-zero remainder, 8210 is not a perfect square.
  7. The current quotient is 90. The next integer is 91. Square 91: 91² = 8281.
  8. Subtract the original number from this perfect square: 8281 - 8210 = 71.

Therefore, the least number to be added to 8210 to make it a perfect square is 71.

Adding 71 to 8210 gives 8281, which is the perfect square of 91.

In conclusion, finding the least number to subtract from or add to a given number to make it a perfect square is a fundamental concept in number theory. Through the methodical application of the long division method and careful analysis of remainders and differences, we can efficiently solve these problems. The examples provided in this article serve as a comprehensive guide to understanding and applying these techniques. Mastery of these concepts not only enhances mathematical proficiency but also provides a solid foundation for tackling more complex problems in various mathematical domains. Whether you are a student or a math enthusiast, understanding these methods will undoubtedly enrich your problem-solving toolkit.