What is the least number by which 16800 must be divided to get a number which is a perfect square?

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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Answer

Verified

Hint:We will use a division method to find the square root. Make a pair of two digits from the right to left of $16800$. Check that the quotient should be less than the remainder for perfect square. Subtract the obtained remainder from the given number to make it a perfect square.

Complete step by step solution:

We know that to find the least number that needs to be subtracted from the given number to make it perfect square, use the long division method. To find the square root of the given number by using a long division method, we need to make a pair of two digits from right to left as $\overline 1 \;\overline {68} \;\overline {00} $.

We get that the remainder is $159$ and the quotient is $129$.The square of obtained quotient is:${\left( {129} \right)^2} = 16641$From the above result we can say that the square of $129$ is a number less that is $159$ less than the given number $16800$.We will subtract $159$ from $16800$ to get a perfect square number.The least number that should be subtracted from $16800$ to make it a perfect square is $159$.

Therefore, the option (B) is correct.

Note:

We can use an alternative method to use the number that should be subtracted from the given number to make it perfect square. We need to find a number whose square is greater than 16800 and a number whose square is less than 16800. Now, subtract the square numbers, the resulting number is the required result.Make sure that the remainder should be subtracted from the given number. Do not add the remainder otherwise this may lead to incorrect answers.