Perfect Square or Square Number

What are called perfect square or square number?

Natural numbers that are squares of other natural numbers are called perfect square or square number.

For example;

We know that; 1 = 1²; 4 = 2²; 9 = 3²; 16 = 4²; 25 = 5² and so on.

Thus 1, 4, 9, 16, 25, etc., are perfect squares.

To find out if the given number is a perfect square:

If the prime factors of a number are grouped in pairs of equal factors, then that number is called a perfect square. Or, in other words if a perfect square number is always expressible as the product of pairs of equal factors.

1. Find out if the following numbers are perfect squares:

(i) 144     (ii) 90     (iii) 180

(i) 144

Resolving 144 into prime factors, we get

144 = 2 × 2 × 2 × 2 × 3 × 3

(grouping the factors into the pairs of equal factors)

Therefore, 144 is a perfect square.

(ii) 90

Resolving 90 into prime factors, we get

90 = 2 × 3 × 3 × 5

(Here 3 is grouped in pairs of equal factors and 2 and 5 are not grouped in pairs of equal factors)

Therefore, 90 is not a perfect square.

(iii) 180

Resolving 180 into prime factors, we get

180 = 2 × 2 × 3 × 3 × 5

(Here 2 and 3 are grouped in pairs of equal factors and 5 is not grouped in pairs of equal factors)

Therefore, 180 is not a perfect square.

2. Is 36 a perfect square? If so, find the number whose square is 36.

Solution:

Resolving 36 into prime factors, we get

36 = 2 × 2 × 3 × 3.

Thus, 36 can be expressed as a product of pairs of equal factors.

Therefore, 36 is a perfect square.

Also, 36 = (2 × 3) × (2 × 3) = (6 × 6) = 6²

Hence, 6 is the number whose square is 36.

3. Is 196 a perfect square? If so, find the number whose square is 196.

Solution:

Resolving 196 into prime factors, we get

196 = 2 x 2 x 7 x 7.

Thus, 196 can be expressed as a product of pairs of equal factors.

Therefore, 196 is a perfect square.

Also, 196 = (2 x 7) x (2 x 7) = (14 x 14) = (14)².

Hence, 14 is the number whose square is 196.

4. Show that 200 is not a perfect square.

Solution:

Resolving 200 into prime factors, we get

200 =2 x 2 x 2 x 5 x 5.

Making pairs of equal factors, we find that 2 is left.

Hence, 200 is not a perfect square.

5. Find the smallest number by which 252 must be multiplied to make it a perfect square.

Solution:

252 = 2 × 2 × 3 × 3 × 7

We observe that 2 and 3 are grouped in pairs and 7 is left unpaired.

If we multiply 252 by the factor 7 then,

252 × 7 = 2 × 2 × 3 × 3 × 7 × 7

1764 = 2 × 2 × 3 × 3 × 7 × 7, which is a perfect square.

Therefore, the required smallest number is 7.

6. Find the smallest number by which 396 must be divided so as to get a perfect square.

Solution:

396 = 2 × 2 × 3 × 3 × 11

We observe that 2 and 3 are grouped in pairs and 11 is left unpaired.

If we divide 396 by the factor 11 then,

396 ÷ 11 = (2 × 2 × 3 × 3 × 1̶1̶)/1̶1̶

= 2 × 2 × 3 × 3 = 36, which is a perfect square.

Therefore, the required smallest number is 11.

Square

Square

Perfect Square or Square Number

Properties of Perfect Squares

Square - Worksheets

Worksheet on Squares

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Recent Articles

1. Comparison of Three-digit Numbers | Arrange 3-digit Numbers |Questions

Sep 13, 24 02:48 AM

What are the rules for the comparison of three-digit numbers? (i) The numbers having less than three digits are always smaller than the numbers having three digits as:

2. Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 13, 24 02:23 AM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…

3. 2nd Grade Place Value | Definition | Explanation | Examples |Worksheet

Sep 13, 24 01:20 AM

The value of a digit in a given number depends on its place or position in the number. This value is called its place value.

4. Comparison of Two-digit Numbers | Arrange 2-digit Numbers | Examples

Sep 12, 24 03:07 PM

What are the rules for the comparison of two-digit numbers? We know that a two-digit number is always greater than a single digit number. But, when both the numbers are two-digit numbers
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}