Finding Square Root through Prime Factorisation

Q.

By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number.
(i) $3675$
(ii) $2156$
(iii) $3332$
(iv) $2925$
(v) $9075$
(vi) $7623$
(vii) $3380$
(viii) $2475$

Q.

The least number which is a perfect square and is divisible by each of $16,20and24$ is:

Q. Simplify
$\frac{\sqrt{32}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}=?$

1. $\sqrt{2}$
2. $4$
3. $8$
4. $2$

Q.

Question 131

Solve the following question:

Find the smallest square number divisible by each of the numbers 8, 9 and 10.

Q.

The students of Class VIII of a school donated \(Rs.~2401\) 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.

Q. Find the least square number which is exactly divisible by each of the numbers 6, 9, 15 and 20.

Q.

Find the value of $a$ in the following: $\frac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-\frac{19}{11}$

1. $\frac{9}{11}$

2. $\frac{2}{13}$

3. $\frac{9}{42}$

4. $\frac{11}{9}$

Q.

Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number:
(i) 1225
(ii) 2601
(iii) 5929
(iv) 7056
(v) 8281

Q.

A school collected Rs. 2304 as fees from its students. If each student paid as many paise as there were students in the school, how many students were there in the school ?

Q.

Using the prime factorisation method, find which of the following numbers are perfect squares:
(i) 441
(ii) 576
(iii) 11025
(iv) 1176
(v) 5625
(vi) 9075
(vii) 4225
(viii) 1089

Q.

The smallest number by which 3645 must be multiplied so that it becomes perfect square is

1. 3

2. 15

3. 5

4. 9

Q. The students of a class arranged a picnic. Each student contributed as many rupees as the number of students in the class. If the total contribution is Rs. 1156, find the strength of the class.

Q.

Find the smallest number by which 180 must be multiplied so that it becomes a perfect square. Also, find the square root of the perfect square so obtained.

Q.

Find the square root of each of the following by prime factorization.
(i) 441 (ii) 196
(iii) 529 (iv) 1764
(v) 1156 (vi) 4096
(vii) 7056 (viii) 8281
(ix) 11664 (x) 47089
(xi) 24336 (xii) 190696
(xiii) 586756 (xiv) 27225
(xv)3013696

Q.

The square root of 529 is ___ . (Use division method to solve)

1. Cannot be determined
2. 32
3. 33
4. 23

Q.

Multiply $3\sqrt{28}\text{by}2\sqrt{7}$.

Q.

A PT teacher wants to arrange maximum possible number of 6000 students in a field such that the number of rows is equal to the number of rows is equal to the number of columns. Find the number of rows if 71 were left out after arrangement.

Q. Find the smallest number by which 2925 must be divided to obtain a perfect square. Also, find the square root of the perfect square so obtained.

Q. Find the square root of each of the following numbers by using the method of prime factorisation:
(1) 225
(2) 441
(3) 729
(4) 1296
(5) 2025
(6) 4096
(7) 7056
(8) 8100
(9) 9216
(10) 11025
(11) 15876
(12) 17424

Q.

Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the number whose square is the resulting number.

Q. 1225 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.

Q.

Find the smallest number by which the given number must be multiplied so hat the product is a perfect square.
(i) 23805 (ii) 12150 (iii) 7688

Q.

Question 109

Solve the following question:

Find the least square number, which is exactly divisible by 3, 4, 5, 6 and 8.

Q.

By what numbers should each of the following be divided to get a perfect square in each case? Also find the number whose square is the new number?
(i) 16562 (ii) 3698
(iii) 5103 (iv) 3174
(v) 1575

Q.

By what number should each of the following numbers be multiplied to get a perfect square in each case ? Also, find the number whose square is the new number.
(i) 8820 (ii) 3675 (iii) 605 (iv) 2880 (v) 4056 (vi) 3468 (vii) 7776

Q.

Using prime factorization find the square root of $4761$

Q.

Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect square.

Q.

Find the square root of 2025 by prime factorisation.

1. 45

2. 35

3. 25

4. 50

Q. Evaluate: $\frac{\sqrt{1183}}{\sqrt{2023}}$

Q.

Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.