What is the smallest number by which 16800 must be divided so that the quotient is a perfect cube?

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✨ [Answered] What is the least number by which 16,800 must be divided to get a number which is perfect square

42, 12, 21, 40 are given options. Correct answer is 42.

Question

What is the least number by which 16,800 must be divided to get a number which is perfect square

What is the least number by which 16,800 must be divided to get a number which is perfect square Answer is"42"

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🌐 Andhra Pradesh answer please

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स्रोत : foxoyo.com

The least number to be subtracted from 16800 to make it a perfect square, is

Click here👆to get an answer to your question ✍️ The least number to be subtracted from 16800 to make it a perfect square, is

Question

The least number to be subtracted from 16800 to make it a perfect square, is

249

159

169

219

Easy Open in App Solution Verified by Toppr

Correct option is B)

This shows that (129)² is less than 16800 by 159, So the least number to be subtracted from 16800 is 159.

Required perfect Square number= 16800- 159= 16641

√16641 = 129

Hence , the required this number is 159.

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स्रोत : www.toppr.com

What is the least number by which 16 800 must be divided to get a number perfect square a 21

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The following steps will be useful to find the least number by which the given number has to be divided to get a perfect square.

Step 1 :

Decompose the given numbers into its prime factors.

Step 2 :

Write the prime factors as pairs such that each pair has two same prime factors.

Step 3 :

Find the prime factor which does not occur in pair. That is the least number by which the given number has to be divided to get a perfect square.

Example 1 :

Find the least number multiplied by 200 to get a perfect square.

Solution :

Decompose 200 into its prime factors.

Prime factors of 200 :

200 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5

= (2 ⋅ 2) ⋅ 2 ⋅ (5 ⋅ 5)

The prime factor 2 does not occur in pair.

So, '2' is the least number by which 200  has to be divided to get a perfect square.

Justification :

√(200/2) = √[(2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5)/2]

√100 = √(2 ⋅ 2 ⋅ 5 ⋅ 5)

= 2 ⋅ 5

= 10

Further,

200/2 = 100 = 102

Example 2 :

Find the least number by which 252 has to be divided to get a perfect square.

Solution :

Decompose 252 into its prime factors.

Prime factors of 252 :

252 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7

= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 7

The prime factor 7 does not occur in pair.

So, '7' is the least number by which 252 has to divided to get a perfect square.

Justification :

√(252/7) = √[(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7)/7]

√36 = √(2 ⋅ 2 ⋅ 3 ⋅ 3)

= 2 ⋅ 3

= 6

Further,

252/7 = 36 = 62

Example 3 :

Find the least number multiplied by 1620 to get a perfect square.

Solution :

Decompose 1620 into its prime factors.

Prime factors of 1620 :

1620 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ (3 ⋅ 3) ⋅ 5

The prime factor 5 does not occur in pair.

So, '5' is the least number to be multiplied by 180 to get a perfect square.

Justification :

√(1620/5) = √[(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5)/5]

√324 = √(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

= 2 ⋅ 3 ⋅ 3

= 18

Further,

1620/5 = 324 = 182

Example 4 :

Find the least number multiplied by 2925 to get a perfect square.

Solution :

Decompose 2925 into its prime factors.

Prime factors of 2925 :

2925 = 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 13

= (3 ⋅ 3) ⋅ (3 ⋅ 3) ⋅ 13

The prime factor 13 does not occur in pair.

So, '13' is the least number by which 2925 has to be divided to get a perfect square.

Justification :

√(2925/13) = √[(3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 13)/13]

√225 = √(3 ⋅ 3 ⋅ 5 ⋅ 5)

= 3 ⋅ 5

= 15

Further,

2925/5 = 225 = 152

Example 5 :

Find the least number multiplied by 384 to get a perfect square.

Solution :

Decompose 384 into its prime factors.

Prime factors of 384 :

384 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3

= (2 ⋅ 2) ⋅ (2 ⋅ 2) ⋅ (2 ⋅ 2) ⋅ 2 ⋅ 3

The prime factors 2 and 3 do not occur in pair.

Product of 2 and 3 :

2 ⋅ 3 = 6

So, '6' is the least number by which 384 has to be divided to get a perfect square.

Justification :

√(384/6) = √[2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3)/6]

√64 = √[2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3)/(2 ⋅ 3)]

= √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

= 2 ⋅ 2 ⋅ 2

= 8

Further,

384/6 = 64 = 82

Example 6 :

Find the least number by which 120 has to be divided to get a perfect square.

Solution :

Decompose 120 into its prime factors.

Prime factors of 120 :

120 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ 2 ⋅ 3 ⋅ 5

The prime factors 2, 3 and 5 do not occur in pair.

Product of 2, 3 and 5 :

2 ⋅ 3 ⋅ 5 = 30

So, '30' is the least number by which 120 has to be divided to get a perfect square.

Justification :

√(120/30) = √[(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)/30]

√4 = √[(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)/(2 ⋅ 3 ⋅ 5)]

= √(2 ⋅ 2)

= 4

Further,

120/30 = 4 = 22

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