Olympiad Test: Square And Square Roots - Class 8 MCQ

Given 

The greatest number of five digits

Hence 59 must be subtracted.

∴ 384 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
so it lacks a 2 and 3

∴ Required number = 6

Consider option (b)

Here 
Clearly 1024 is the smallest number of four digits which is a perfect square.

Greatest number of four digits = 9999

2292 + 12 = 2304 which is a perfect square.

To determine the smallest number by which 557568 must be divided so that it becomes a perfect square, let's factorize 557568 and see the prime factors:

Step 1: Factorize 557568

We can start by dividing 557568 by the smallest prime numbers:

557568÷2=278784(Divide by 2)

278784÷2=139392(Divide by 2)

139392÷2=69696(Divide by 2)

69696÷2=34848(Divide by 2)

34848÷2=17424(Divide by 2)

17424 ÷ 2=8712(Divide by 2)

8712 ÷ 2 = 4356(Divide by 2)

4356÷2=2178(Divide by 2)

2178÷2=1089(Divide by 2, but can’t divide further, stop here with 1089)

To make this exponent even, we must divide by 2 to reduce the exponent to 8 (which is even). Hence, the smallest number by which 557568 should be divided to make it a perfect square is 2.

Answer: 4. 2

396 = 2 × 2× 3 × 3 × 11
To make a perfect square, 11 is multiplied to 396.

6241 – 6203 = 38
38 must be added to 6203 to make a perfect square.

Here LCM of 8, 15, 20 = 120. and
120 = 2  × 2 × 2 × 3 × 5
To make 120 a perfect square, it must be multiplied by 2 × 3 ×5 = 30
120 × 30 = 3600

36 × 36 = 1296
37 × 3 7 = 1369
69 must be added to make 1300 as perfect square.

Perimeter of an equilateral triangle
= 4 + 4 + 4 = 12 cm

Perimeter of square = 12 cm ⇒ 4 × side = 12 ⇒ Side =  3 cm.
∴    Length of diagonal = 

Let the numbers are x and 3x.

∴
 

Let breadth  be x m
∴ Length = 3x m
Now
 
∴ 
Perimeter = 2 (10.77 + 32.31) = 86.16 m

Here 3² + 4² = 5² 
         6² + 8² = 10²
    2² + 3² ≠ 4²
    12² + 35² = 37²
∴ (c) is not a Pythagorean triplet.

Speed of cycle = 18 km/hour
18 x 5/18
5m/sec
Area of square = 60025
Side of square =  = 245 m
Perimeter of square = 245 × 4 m
∴    Time taken = 

Area of square = Area of rectangle

∴ Side of square =  = 6.8 m

Perimeter of square = 76
⇒    4 × side = 76 ⇒ side = 76/4 = 19
∴ Area of square = 19² = 361 m²

Here 729 = 27², 324 = 18², 441 =  21², 625 = 25²
or
Square of an odd number is an odd number.