Is 2352 a perfect square? If not, find the smallest number by which 2352 must be multiplied so that the product is a perfect square. Find the square of the new number.
Answer
Hint: In the above question, a whole number is given. We have to find out if the number is a perfect square or not. Now, we know that this number is not a perfect square. Now, we have to find a smallest number which can be multiplied to it so that the number becomes a perfect square. Firstly, we will factorize the number given and then look for a number which can be multiplied to make this number a perfect square.
Complete step-by-step solution:
In the above
question, we are given with whole number 2352
Now, finding the square root of this number,
We get,
$ \Rightarrow \sqrt {2352} = 48.497$
Now, the square root of the number $48.497$ . So, we can say that the number is not a perfect square.
Now, we have to find a smallest number which we can multiply to the number to make it a perfect square.
Firstly, we will do the factorization of the number 2352.
Now, the factor of 2352 is $2 \times 2 \times 2 \times 2 \times 3 \times 7
\times 7$ . Now, we know that the number can only be a perfect square when the factors of the number are in pairs.
So, from the factors of the number 2352, we can see that all the factors are in pairs, except factor 3.
So, we will multiply the number 2352 with 3 to make it a perfect square.
Now, to verify that 3 is the smallest number which is multiplied to 2352 to make it a perfect square, we need to multiply 3 with the number and then find its square root.
$ \Rightarrow 3 \times
2352 = 7056$
Now, finding the square root of the number $7056$ ,
We get,
$ \Rightarrow \sqrt {7056} = 84$
Hence, the number is a perfect square and the perfect square is 84.
Note: For a number to be called as a perfect square then the square root of that number should be the whole number. If the square root is not a whole number then it will be a terminating or non terminating decimal.
There are simple tricks or ideas to find out squares and square root
easily
For example - if there is any number whose unit digit or at ones place there is 5 then its square or square root will contain 5 at its ones place .
It can be observed that 225 is square root of 15
Again , we know that 25 is a square of 5.
Jump to
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3
- Exercise 6.4
- Rational Numbers
- Linear Equations in One Variable
- Understanding Quadrilaterals
- Practical Geometry
- Data Handling
- Squares and Square Roots
- Cubes and Cube Roots
- Comparing Quantities
- Algebraic Expressions and Identities
- Visualising Solid Shapes
- Mensuration
- Exponents and Powers
- Direct and Inverse Proportions
- Factorisation
- Introduction to Graphs
- Playing with Numbers
NCERT Solutions Class 8 Mathematics Solutions for Exercise 6.3 in Chapter 6 - Squares and Square Roots
Q10] Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.
Answer:
Solution:
L.C.M. of 8, 15 and 20 is 120.
Prime factors of 120 = 2 x 2 x 2 x 3 x 5
Here, prime factors 2, 3 and 5 have no pair. Therefore 120 must be multiplied by 2 x 3 x 5 to make it a perfect square.
\therefore120\times2\times3\times5=3600
Hence, the smallest square number which is divisible by 8, 15 and 20 is 3600.
Video transcript
hello kids welcome to Lido Homework in this video we're going to solve this A question which says to find the smallest square number that is divisible by each of the numbers 8 15 and 20. so to find the smallest square number Divisible by 8 15 and 20 Let's find out the lcm of the given numbers 8 15 and 20. okay so let's write that down 8 15 20. two fours are eight-fifteen two tens are two twos are four fifteen as it is two fives are ten two ones are two fifteen and five as it is the next number would be three three fives are fifteen and five as it is and the next number would be five okay so the lcm is 2 into 2 into 2 into 3 into 5 this is 8 into 3 into 5 which will give you 120 so the lcm of 8 15 and 20 is 120. Now pay attention to the word square that's mentioned over here you're just not finding any smallest number that not the smallest number that is there but you find the smallest square number so for that we need to figure out if 120 is a perfect square or not so for that we'll have to check it with prime factorization let's prime factorize 120 two six are zero two threes are two are two fives are three fives are and five ones Okay, so the factors in the prime factors of 120 are 2 into 2 into 2 into 3 into 5. so when you have to find a perfect square we first group the numbers and see how many numbers don't have a group and we multiply by those numbers right so you can see here 2 has a group great now this 2 is single 3 is single and 5 is single they don't have a group so what i have to do is I'll have to multiply 120 by one more two one more three and one more five so what What I'm doing is I'm going to add one more two here, one more three here. and another five here so that i get complete groups right and then the number that i get would be a perfect square so 120 into 2 into 3 into 5 will give me 3 6 double 0. so the smallest the square number that is divisible by 8 15 and 20 is six hundred i hope i was able to uh solve your doubt with this particular video please feel free to like to share and comment and I'll see you in another video Thank you.
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