What is the least number by which 343 must be multiplied with in order to get a perfect square?

343 = (7*7)*7

Since 7 does not form a pair we multiply 343 with 7 i.e.

343*7 = 2401 = (7*7)(7*7) = 492

Thus square root of 2401 is 49.

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

Kindly mail your feedback to

We always appreciate your feedback.

For StudentsFor ParentsFor SchoolsSuccess StoriesOutcomesAI

'I Did It' Mathematics>Squares and Square Roots>Exercise 2.1>Q 9

1. Perfect Square:

A natural number x is a perfect square if there exists a natural number y such that x=y2. In other words, a natural number x is a perfect square, if it is equal to the product of a number with itself.

2. Properties of Squares Numbers:

(i) A number ending in 2, 3, 7, or 8 is never a perfect square.

(ii) The number of zeroes in the end of a perfect square is never odd. So, a number ending in an odd number of zeroes is never a perfect square.

(iii) Squares of even numbers are always even.

(iv) Squares of odd numbers are always odd.

3. General Properties of Perfect Squares:

(i) For any natural number n, we have n2= (Sum of first n odd natural numbers)

(ii) The square of a natural number, other than 1, is either a multiple of 3 or exceeds a multiple of 3 by 1.

(iii) The square of a natural number, other than 1, is either a multiple of 4 or exceeds a multiple of 4 by 1.

(iv) There are no natural numbers p and q such that p2=2q2

4. Pythagorean Triplets:

For any natural number n greater than 1, (2n, n2−1, n2+1), is a Pythagorean triplet.

5. Square roots:

The square root of a given natural number n is that natural number which when multiplied by itself gives n as the product and we denote the square root of n by n. Thus, n=m⇔n=m2.

6. Finding Square Roots:

(i) In order to find the square root of a perfect square, resolve it into prime factors; make pairs of similar factors and take the product of prime factors, choosing one out of every pair.

(ii) For finding the square root of a decimal fraction, make the even number of decimal places by affixing a zero, if necessary; mark off periods and extract the square root; putting the decimal point in the square root as soon as the integral part is exhausted.

7. Properties of Square Roots:

For positive numbers a and b, we have

(i) ab=a×b

(ii) ab=ab

How to figure out the nearest square root of a non-perfect square number or guessing the square root of a perfect square number? Let's learn in this video.

Do you know the tedious procedure of finding the square root of decimals easier? You can do that with the help of the long division method! Watch the video t...

You know that if there is a point in between two numbers, it is a decimal number. But how do you find the square root of decimal numbers using the prime fact...

Boost your brain quickly & test your analytical skills with our incredible math puzzle! This will let you know more about the perfect square numbers. So let’s explore this video to know more!

The long division method in finding the square root seems to be a lengthy method. Is there any shortcut to find the square root of numbers? Watch this video...

Sometimes, we have to do too many things at once, and one of them could be finding squares. Watch this video, and find time to calculate those squares, you k...