What is the smallest number that can be multiplied to 297 to make it a perfect square number

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

1. Decompose the given numbers into its prime factors.

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

3. Find the prime factor which does not occur in pair. That is the least number to be multiplied by the given number 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 to be multiplied by 200 to get a perfect square.

Justification :

√[2(200)] = √[2(2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5)]

√400 = √[(2 ⋅ 2)(2 ⋅ 2)(5 ⋅ 5)]

= 2 ⋅ 2 ⋅ 5

= 20

Further,

2(200) = 400 = 202

Example 2 :

Find the least number multiplied by 252 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 to be multiplied by 252 to get a perfect square.

Justification :

√[7(252)] = √[7(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7)]

√1764 = √[(2 ⋅ 2)(3 ⋅ 3)(7 ⋅ 7)]

= 2 ⋅ 3 ⋅ 7

= 42

Further,

7(252) = 1764 = 422

Example 3 :

Find the least number multiplied by 180 to get a perfect square.

Solution :

Decompose 180 into its prime factors.

Prime factors of 180 :

180 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ (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 :

√[5(180)] = √[5(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5)]

√900 = √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 3 ⋅ 5

= 30

Further,

5(180) = 900 = 302

Example 4 :

Find the least number multiplied by 90 to get a perfect square.

Solution :

Decompose 90 into its prime factors.

Prime factors of 90 :

90 = 2 ⋅ 3 ⋅ 3 ⋅ 5

= 2 ⋅ (3 ⋅ 3) ⋅ 5

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

Product of 2 and 5 :

2 ⋅ 5 = 10

So, '10' is the least number to be multiplied by 90 to get a perfect square.

Justification :

√[10(90)] = √[10(2 ⋅ 3 ⋅ 3 ⋅ 5)]

√900 = √[(2 ⋅ 5)(2 ⋅ 3 ⋅ 3 ⋅ 5)]

= √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 3 ⋅ 5

= 30

Further,

10(90) = 900 = 302

Example 5 :

Find the least number multiplied by 120 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 to be multiplied by 120 to get a perfect square.

Justification :

√[30(120)] = √[30(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]

√3600 = √[(2 ⋅ 3 ⋅ 5)(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]

= √[(2 ⋅ 2)(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 2 ⋅ 3 ⋅ 5

= 60

Further,

30(120) = 3600 = 602

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Solution:

(i) 252 = 2 x 2 x 3 x 3 x 7

Here, prime factor 7 has no pair. Therefore 252 must be multiplied by 7 to make it a perfect square.

\therefore252\times7=1764

And (i) \sqrt{1764}=2\times3\times7=42

(ii) 180 = 2 x 2 x 3 x 3 x 5

Here, prime factor 5 has no pair. Therefore 180 must be multiplied by 5 to make it a perfect square.

\therefore180\times5=900

And \sqrt{900}=2\times3\times5=30

(iii) 1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7

Here, prime factor 7 has no pair. Therefore 1008 must be multiplied by 7 to make it a perfect square.

\therefore1008\times7=7056

And \sqrt{7056}=2\times2\times3\times7=84

(iv) 2028 = 2 x 2 x 3 x 13 x 13

Here, prime factor 3 has no pair. Therefore 2028 must be multiplied by 3 to make it a perfect square.

\therefore2028\times3=6084

And \sqrt{6084}=2\times2\times3\times3\times13\times13=78

(v) 1458 = 2 x 3 x 3 x 3 x 3 x 3 x 3

Here, prime factor 2 has no pair. Therefore 1458 must be multiplied by 2 to make it a perfect square.

\therefore1458\times2=2916

And \sqrt{2916}=2\times3\times3\times3=54

(vi) 768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Here, prime factor 3 has no pair. Tehrefore 768 must be multiplied by 3 to make it a perfect square.

\therefore768\times3=2304

And \sqrt{2304}=2\times2\times2\times2\times3=48