1. Square root
Least number
Multiply & Divide

2. Example 1:
Find the smallest number by which 180 must be multiplied so that the product becomes a perfect square.
Solution:
By Prime factorization method, we get
By pairing the factors we get
We notice 5 needs a pair
so we multiply both sides by 5
180 = 2 x 2 x 3 x 3 x 5
180 x 5 = 2 x 2 x 3 x 3 x 5 x 5
Hence, the smallest number to be multiplied with 180 is 5.
Incase of multiplication, multiply the number that needs a pair

3. Example 2:
Find the smallest number by which 360 must be multiplied so that the product becomes a perfect square.
Solution:
By Prime factorization method, we get
By pairing the factors we get
We notice 2 and 5 need a pair
so we multiply both sides by 2 x 5
360 = 2 x 2 x 2 x 3 x 3 x 5
360 x 2 x 5 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5
Hence, the smallest number to be multiplied with 360 is 2 x 5 =10.
Incase of multiplication, multiply the number that needs a pair

4. Example 3:
Find the smallest number by which 1100 must be divided so that the quotient becomes a perfect square.
Solution:
By Prime factorization method, we get
By pairing the factors we get
We notice 11 is without pair
so we divide both sides by 11 to remove it
1100 = 2 x 2 x 5 x 5 x 11
= 2 x 2 x 5 x 5 x 11 11
Hence, the smallest number to be divided with 1100 is 11.
Incase of division, divide the number that is without a pair

5. Example 3:
Find the smallest number by which 1152 must be divided so that the quotient becomes a perfect square.
Solution:
By Prime factorization method, we get
By pairing the factors we get
We notice 2 is without pair
so we divide both sides by 2 to remove it
1152 = 2 × 2 × 2 × 2 × 2 x 2 x 2 x 3 x 3
= 2 × 2 × 2 × 2 × 2 x 2 x 2 x 3 x 3 2
Hence, the smallest number to be divided with 1152 is 2.
Incase of division, divide the number that is without a pair

6. Try These:
Find the smallest number by which 2880 must be multiplied so that the product becomes a perfect square.
2. Find the smallest number by which 1575 must be divided so that the quotient is a perfect square.