By what least number should the given number be multiplied to get a perfect square number?

Solution:

1. Resolving 3675 into prime factors:

$3675=3 \times 5 \times 5 \times 7 \times 7$

Thus, to get a perfect square, the given number should be multiplied by 3.

New number $=\left(3^{2} \times 5^{2} \times 7^{2}\right)=(3 \times 5 \times 7)^{2}=(105)^{2}$

Hence, the new number is the square of 105.

2. Resolving 2156 into prime factors:

$2156=2 \times 2 \times 7 \times 7 \times 11=\left(2^{2} \times 7^{2} \times 11\right)$

Thus to get a perfect square, the given number should be multiplied by 11.

New number $=\left(2^{2} \times 7^{2} \times 11^{2}\right)=(2 \times 7 \times 11)^{2}=(154)^{2}$

Hence, the new number is the square of 154.

3. Resolving 3332 into prime factors:

$3332=2 \times 2 \times 7 \times 7 \times 17=2^{2} \times 7^{2} \times 17$

Thus, to get a perfect square, the given number should be multiplied by 17.

New number $=\left(2^{2} \times 7^{2} \times 17^{2}\right)=(2 \times 7 \times 17)^{2}=(238)^{2}$

Hence, the new number is the square of 238.

4. Resolving 2925 into prime factors:

$2925=3 \times 3 \times 5 \times 5 \times 13=3^{2} \times 5^{2} \times 13$

Thus, to get a perfect square, the given number should be multiplied by 13.

New number $=\left(3^{2} \times 5^{2} \times 13^{2}\right)=(3 \times 5 \times 13)^{2}=(195)^{2}$

Hence, the number whose square is the new number is 195.

5. Resolving 9075 into prime factors:

$9075=3 \times 5 \times 5 \times 11 \times 11=3 \times 5^{2} \times 11^{2}$

Thus, to get a perfect square, the given number should be multiplied by 3.

New number $=\left(3^{2} \times 5^{2} \times 11^{2}\right)=(3 \times 5 \times 11)^{2}=(165)^{2}$

Hence, the new number is the square of 165.

6. Resolving 7623 into prime factors:

$7623=3 \times 3 \times 7 \times 11 \times 11=3^{2} \times 7 \times 11^{2}$

Thus, to get a perfect square, the given number should be multiplied by 7.

New number $=\left(3^{2} \times 7^{2} \times 11^{2}\right)=(3 \times 7 \times 11)^{2}=(231)^{2}$

Hence, the number whose square is the new number is 231.

7. Resolving 3380 into prime factors:

$3380=2 \times 2 \times 5 \times 13 \times 13=2^{2} \times 5 \times 13^{2}$

Thus, to get a perfect square, the given number should be multiplied by 5.

New number $=\left(2^{2} \times 5^{2} \times 13^{2}\right)=(2 \times 5 \times 13)^{2}=(130)^{2}$

Hence, the new number is the square of 130.

8. Resolving 2475 into prime factors:

$2475=3 \times 3 \times 5 \times 5 \times 11=3^{2} \times 5^{2} \times 11$

Thus, to get a perfect square, the given number should be multiplied by 11.

New number $=\left(3^{2} \times 5^{2} \times 11^{2}\right)=(3 \times 5 \times 11)^{2}=(165)^{2}$

Hence, the new number is the square of 165.