FIND THE LEAST NUMBER TO BE MULTIPLIED TO GET A PERFECT SQUARE

Problem 1 :

The least number by which 294 must be multiplied to make it a perfect square, is

a)   2        b)   3        c)   6    d)   24

Solution :

From the questions, it is clear 294 is not a perfect square. To make 294 as perfect square, we will multiply some numerical value or values.

Prime factorization of 294 is,

294 = 7 × 7 × 2 × 3

Here we need one more 2 and one more 3.

To make it a perfect square, it must be multiplied by 2 × 3 = 6

Required number = 6

So, option (c) is correct.

Problem 2 :

Find the smallest number by which 5808 should be multiplied so that the product becomes a perfect square.

a)   2      b)   3       c)   7        d)   11

Solution :

Prime factorization of 5808 is,

5808 = 2 × 2 × 2 × 2 × 3 × 11 × 11

Therefore, when 5808 is multiplied by 3, then it will be perfect square number.

So, option (b) is correct.

Problem 3 :

By which least number should 22050 be multiplied such that the result is a perfect square.

Solution :

Decomposing 22050,

22050 = 5 x 5 x 2 x 21 x 21

To make it as perfect square, we need one more 2. So, the required number is 2.

Problem 4 :

Find the smallest common multiple of 48, 72 and 32 that is a perfect cube.

Solution :

Decomposing 48, 72 and 32, we get

48 = 2 x 2 x 2 x 2 x 3 ==> 2⁴ x 3

72 = 2 x 2 x 2 x 3 x 3 ==> 2³ x 3²

32 = 2 x 2 x 2 x 2 x 2 ==> 2⁵

LCM = 2⁵ x 3²

= 288

Decomposing 288, we get

= 2⁵ x 3²

that is, 2 x 2 x 2 x 2 x 2 x 3 x 3

To make it as perfect cube, we need one more 2 and one more 3.

So, the required number is 2 x 3 ==> 6.

Problem 5 :

What is the smallest common multiple of 72 and 108 that is a perfect square ?

Solution :

Least common multiple of 72 and 108.

72 = 2 x 2 x 2 x 3 x 3 ==> 2³ x 3²

108 = 2 x 2 x 3 x 3 x 3 ==> 2² x 3²

= 2³ x 3²

= 72

Now decomposing 72, we get

= 2³ x 3²

To make it as perfect square

= 2³ x 3² x 2 x 3

So, the required number is 6.

Problem 6 :

By which least number should 4375 be multiplied such that the result is a perfect square ?

Solution :

Decomposing 4375, we get

4375 = 5 x 5 x 5 x 5 x 7

To make it as perfect square, we need one more 7.

4375 = 5 x 5 x 5 x 5 x 7 x 7

So, the required number is 7.

Problem 7 :

By which least number should 72000 be multiplied such that the result is a perfect square ?

Solution :

Decomposing 72000, we get

72000 = 2⁵ x 5² x 3²

To make it as perfect square, we need one more 2.

So, the required number is 2.

Problem 8 :

Find the smallest number by which 375 must be multiplied to obtain a perfect cube.

Solution :

Decomposing 375, we get

375 = 5 x 5 x 5 x 3

= 5³ x 3

To get cube of 3, two more 3's should be included. So, 9 should be multiplied by 375 to make it as perfect cube.

Problem 9 :

You have 275 square inches of wrapping paper. Do you have enough wrapping paper to wrap the gift box shown? Explain.

Solution :

Volume of box = 343 cubic inches

Side length is x.

x³ = 343

x =  ³√343

x =  ³√(7 x 7 x 7)

= 7 inches

Total surface area of cube = 6x²

= 6(7)²

= 6(49)

= 294 square inches

To wrap the gift box, we need 294 square inches of paper. Since it is provided 275 square inches of paper, it is enough.

Problem 10 :

A cube-shaped end table has a volume of 216,000 cubic centimeters. Does the end table fit in the corner shown? Justify your answer

Solution :

Volume of cube shaped table = 216,000 cubic centimeters

Let x be the side length of table.

x³ = 216000

x = ³√216000

x = ³√(60 x 60 x 60)

x = 60 cm

Area of square = 6400 cm²

x²  = 6400

x = √6400

x = √(80 x 80)

= 80 cm

Since the side length of square shape is 80 cm and side length of cubic shape table is 60 cm, the table will fit there.