Square questions are provided here, along with their solutions, based on the Class 8 syllabus. They are prepared as per the NCERT (CBSE) guidelines. Solving these questions will help students understand the concept well, and improve their skills with squares of numbers. Learn more about Squares and square roots.
The Square of a given number can be found by multiplying it twice with itself. For example, 2 × 2 = 4, written as 22. The below table shows the square of numbers from 1 to 20.
|
Number |
Square |
Number |
Square |
|---|---|---|---|
|
1 |
1 |
11 |
121 |
|
2 |
4 |
12 |
144 |
|
3 |
9 |
13 |
169 |
|
4 |
16 |
14 |
196 |
|
5 |
25 |
15 |
225 |
|
6 |
36 |
16 |
256 |
|
7 |
49 |
17 |
289 |
|
8 |
64 |
18 |
324 |
|
9 |
81 |
19 |
361 |
|
10 |
100 |
20 |
400 |
Square Questions with Solutions
Here are a few questions based on squares of numbers.
1. Is 3528 a perfect square?
Solution: The prime factorisation of 3528 is
3528 = 2 × 2 × 2 × 3 × 3 × 7 × 7
Pairing off the factors we find one factor left unpaired
Therefore, 3528 is not a perfect square.
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Perfect squares are the numbers that can be expressed as the square of any number. For example, 64, 121, 25, etc. |
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2. What will be the unit digit of squares of the following numbers?
(i) 2387 (ii) 1001 (iii) 252
Solution: By the property of square numbers, the unit digit of a square number is the same as the square of the unit digit of the number to be squared.
(i) Unit digit of square of 2387 is 9
as 7 × 7 = 49
(ii) Unit digit of square of 1001 is 1
as 1 × 1 = 1
(iii) Unit digit of square of 252 is 5
as 5 × 5 = 25
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Properties of Square Numbers
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3. How many non-square numbers lie between 80 and 81?
Solution: Since 80 and 81 are consecutive natural numbers where n = 80 and n + 1 = 81, the number of non-square numbers is 2n, that is, 2 × 80 = 160
4. Find the smallest perfect square divisible by 3, 4, 5 and 6.
Solution: LCM of 3, 4, 5 and 6 is 60
60 = 2 × 2 × 3 × 5 where 3 and 5 are unpaired.
We must multiply 60 by 3 × 5 to get the smallest square number
The smallest perfect square divisible by 3, 4, 5, and 6 = 60 × 3 × 5 = 900
5. Out of 745 students, the maximum is to be arranged in the school field for a P.T. display, such that the number of rows is equal to the number of columns. Find the number of rows if 16 students were left out after the arrangement.
Solution: Total number of students = 745
Now, 272 < 745 < 282
And 745 – 252 = 745 – 729 = 16
Therefore, there are 27 rows formed.
6. Find a Pythagorean triplet whose one member is 28.
Solution: Pythagorean triples are in the form of 2m, m2 – 1, m2 + 1
Let 2m = 28 ⇒ m = 14
then, m2 – 1 = 142 – 1 = 196 -1 = 195
and m2 + 1 = 142 + 1 = 196 + 1 = 197
Therefore, the Pythagorean triples are 28, 195 and 197
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For any m > 0, (2m)2 + (m2 – 1)2 = (m2 + 1)2 where 2m, (m2 – 1) and (m2 + 1) are known as Pythagorean triples. |
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Also Read:
7. Find the smallest number which should be divided by 1620 so as to get the quotient as a perfect square.
Solution: Now, 1620 = 2 × 2 × 3 × 3 × 3 × 3 × 5
Only 5 is left unpaired, so 1620 must be divided by 5, 1620 ÷ 5 = 324 = 2 × 2 × 3 × 3 × 3 × 3 which is a perfect square number.
8. What is the smallest number which should be multiplied by 2028 to make it a perfect square number?
Solution: 2028 = 2 × 2 × 3 × 13 × 13
Only 3 is left unpaired, we must multiply 2028 by 3 to get a perfect square,
Therefore, 2028 × 3 = 6084 is a perfect square.
9. A man plants his orchard with 5625 trees, and arranges them so that there are as many rows as there are trees in each row. How many rows are there?
Solution: Let x be the number of rows. Since there are equal number of columns and rows
Total number of trees = x × x = x2 = 5625
Therefore, x = √5625 = √(5 × 5 × 5 × 5 × 3 × 3) = 5 × 5 × 3 = 75
There are 75 rows.
10. Find the square of 15 ⅔.
Solution: (15 ⅔ .)2 = (47/3)2 = (47)2/32 = 2209/9
Practice Questions
1. Find the smallest number that should be multiplied with 882 to make it a perfect square.
2. Find the smallest possible number that should be divided from 2187 to make it a perfect square number.
3. Find the smallest perfect square number which is divisible by 4, 9 and 10.
4. In a basket there are 50 flowers. A man goes to worship and puts as many flowers in each temple as there are temples in the city. Thus, he needs 8 baskets of flowers. Find the number of temples in the city.
5. Find the Pythagorean triplet whose one member is 12.
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