Worksheet Solutions: Square | Mental Mathematics for Class 8 PDF Download

Q1: Is 3528 a perfect square?
Sol: 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.

Q2: How many non-square numbers lie between 80 and 81?
Sol: 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

Q3: 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.
Sol: Total number of students = 745
Now, 27² < 745 < 28²
And 745 – 25² = 745 – 729 = 16
Therefore, there are 27 rows formed.

Q4: Find the smallest number which should be divided by 1620 so as to get the quotient as a perfect square.
Sol: 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.

Q5: 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?
Sol: 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.

Q6: What will be the unit digit of squares of the following numbers?
(i) 2387
(ii) 1001
(iii) 252
Sol: 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

Q7: Find the smallest perfect square divisible by 3, 4, 5 and 6.
Sol: 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

Q8: Find a Pythagorean triplet whose one member is 28.
Sol: 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

Q9: What is the smallest number which should be multiplied by 2028 to make it a perfect square number?
Sol: 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.

Q10:  Find the square of 15 ⅔.
Sol: (15 ⅔ .)² = (47/3)² = (47)²/32 = 2209/9