Understanding Perfect Squares and Finding the Least Number to Subtract

Perfect squares are a fascinating and essential part of mathematics. A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4 squared (42 16). The problem at hand involves finding the smallest positive integer that must be subtracted from a given number to make it a perfect square. In this article, we aim to explore how to determine this least number through examples and the long division method.

Example Analysis

Example 1: What is the least number which must be subtracted from 549162 to make it a perfect square?

First, we need to find the nearest perfect square below 549162. Let's start by determining the integer whose square is just less than 549162.

(sqrt{549162} approx 740.76)

The integer part of this square root is 740, so we consider (740^2).

Calculating (740^2):

[begin{aligned} 740 740 times 740 547600 741 741 times 741 549081 549162 - 549081 81 end{aligned}]

Thus, the least positive integer that must be subtracted from 549162 to obtain a perfect square is 81.

Examples and Methods

Example: Subtracting to Reach a Perfect Square

Example 2: Determine the least number that should be subtracted from 1900 to get a perfect square.

[begin{aligned} 40^2 1600 43^2 1849 44^2 1936 1849 43^2 1900 - 1849 51 end{aligned}]

The least number to subtract from 1900 to make it a perfect square is 51.

Long Division Method

The long division method is another tool to find the square root of a number. Here's an example to illustrate this method:

Example 3: Find the square root of 15440 using the long division method.

[begin{array}{c|cccc} 124 hline 15440 124 . 2 hline -144 hline 1000 40 hline 44 -440 hline 640 hline -64 hline 0 hline end{array}]

The quotient is 124, and 1242 is 15376. Therefore, the least number to subtract from 15440 to make it a perfect square is 64.

Example 4: The square root of 11.075 is approximately 10.5. Squaring 10.5 gives 110.25. Thus, the least number to subtract from 11075 is 50.

2116 is 462 and 472 is 2209. So, the smallest number that can be subtracted from 2116 to make it a perfect square is 4.

Example 5: The square root of 3799 by the long division method gives a quotient of 61 with a remainder of 78. Subtracting the remainder from 3799 results in 3721, which is a perfect square (612).

Key Concepts

The square root of a number is the value that, when raised to the power of 2, equals the original number. The least number to subtract from a given number to make it a perfect square can be determined by finding the nearest perfect square that is less than or equal to the given number and then subtracting that perfect square from the original number. The long division method can be used to find the square root of a number accurately.

Understanding and mastering these techniques will help in solving problems related to perfect squares efficiently and accurately. Whether it be finding the square root through the long division method or determining the least number to subtract to obtain a perfect square, these methods are essential tools in the toolkit of a math enthusiast or a professional in the field of data analytics and SEO.