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Q.10 Find the smallest square number that is divisible by each of the numbers $8,15$ and $20$.

This has to be done in two steps. First we will find LCM of given numbers, then we will make it perfect square. So the LCM of 8, 15, 20 is 120   .                                  8 = 222  ;           15 = 35   ;             20 =  225 Prime factorisation of 120 gives =  . To make it a perfect square we need to multiply it with 30. So the smallest square number that is divisible by each of the...

Q.9 Find the smallest square number that is divisible by each of the numbers $4,9$  and $10$.

This has to be done in two steps. First, we will find LCM of given numbers, then we will make it a perfect square. So the LCM of 4, 9, 10 is   180.                                  4 = 22  ;           9 = 33   ;             10 =  25 Prime factorisation of 180 gives = . To make it a perfect square we need to multiply it with 5. So, the smallest square number which is divisible by each of the...

Q.8  2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

The total number of plants = No. of rows  No. of plants in 1 row. Since in this case  no.of rows = no. of plants in each row.  Thus let us assume the number of rows to be x.  Then the equation becomes   :         Prime factorization of 2025 gives =    So value of x is = 45. Hence no. of rows = 45; and no. of plants in each row = 45.

Q.7  The students of Class VIII of a school donated $\inline Rs.2401$ in all, for Prime Minister’s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.

Let the number of students in a class be x. According to question, Number of student = money donated by each of the students So total money donated =  or                       Prime factorization of        So the number of students in the class = 49.

Q.6 For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.

(i) 252

(ii) 2925

(iii) 396

(iv) 2645

(v) 2800

(vi) 1620

(i) 252: Prime factorization of 252 gives = .                  For making pairs we will divide the given number by 7.                  The obtained number is 36 and its square root is 6. (ii) 2925: Prime factorization of 2925 gives =                    To make pairs divide the given number by 13.                   So the obtained number is 225 and its square root is 15. (iii) 396: Prime...

Q.5  For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained

(i) 252

(ii) 180

(iii) 1008

(iv) 2028

(v) 1458

(vi) 768

(i) 252 :    Prime factorisation of 252 = .                To make pairs we will multiply 252 with 7.                So the number is 1764 and its square root is 42.   (ii) 180 :     Prime factorisation of 180 = .                  To make it perfect square, multiply by 5.                  So the number is 900 and its square root is 30.   (iii) 1008 :   Prime factorization of 1008 gives = .    ...

Q.4 Find the square roots of the following numbers by the Prime Factorisation Method.

(x) 8100

The solution for the above-written question is as follows We have in 8100. By prime factorization, we get :       or                                                     . So square root of 8100 is 90.

Q.4  Find the square roots of the following numbers by the Prime Factorisation Method.

(ix) 529

The solution for the above-written question is as follows We have 529. Prime factorization gives          So square root of 529 is 23.

Q4. Find the square roots of the following numbers by the Prime Factorisation Method.

(viii) 9216

The solution for the above-written question is as follows prime factorization of 9216,                or           . Thus, the square root of 9216 is 96.

Q.4  Find the square roots of the following numbers by the Prime Factorisation Method.

(vii) 5929

The solution for the above-written question is as follows  Prime factorization of number 5929,               or           . Thus, the square root of 5929 is 77.

Q.4  Find the square roots of the following numbers by the Prime Factorisation Method.

(vi) 9604

We have in 9604. By prime factorization we get,                or           Hence the square root of 9604 is 98.

Q.4 Find the square roots of the following numbers by the Prime Factorisation Method.

(v) 7744

We have in 7744. By prime factorization, we get            or        Thus the square root of 7744 is 44.

Q.4 Find the square roots of the following numbers by the Prime Factorisation Method.

(iv) 4096

We have 4096, by prime factorization:              or          . So the square root of 4096 is 64.

Q4. Find the square roots of the following numbers by the Prime Factorisation Method.

(iii) 1764

We have 1764, by prime factorization we get              or           Thus the square root of 1764 is 42.

Q.4  Find the square roots of the following numbers by the Prime Factorisation Method.

(ii) 400

By prime factorization, we get                    or                Thus the square root of 400 is 20

Q.4 Find the square roots of the following numbers by the Prime Factorisation Method.

(i) 729

By prime factorisation, we know that                   or              Thus the square root of 729 is 27.

Q.3  Find the square roots of $100$ and $169$ by the method of repeated subtraction.

(i)   For 100 :-  100 - 1 = 99       99 - 3 = 96       96 - 5 = 91       91 - 7 = 84       84 - 9 = 75       75 - 11 = 64       64 - 13 = 51          51 - 15 = 36                       36 - 17 = 19       19 - 19 = 0. We obtain zero at 10th step so    (ii)    For 169 :- 169 - 1 = 168         168 - 3 = 165         165 - 5 = 160         160 - 7 = 153         153 - 9 = 144         144 - 11 = 133  ...

Q.2 Without doing any calculation, find the numbers which are surely not perfect squares.

(i) 153

(ii) 257

(iii) 408

(iv) 441

As we know the units place of a perfect square cannot be 2, 3, 7, and 8. So 153,  257,  408 are surely not perfect squares.

Q1. What could be the possible ‘one’s’ digits of the square root of each of the following numbers?

(iv) 657666025

We know that square of a number ending with 5 gives 5 at its units place. So the possible ‘one’s’ digits of the square root of 657666025 are 5.

Q1. What could be the possible ‘one’s’ digits of the square root of each of the following numbers?

(iii) 998001

We know that square of digits ending with 1 and 9 gives 1 at units place. So the possible ‘one’s’ digits of the square root of 998001 are 1 and 9.
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