Question1: find the least perfect squared number that is divisible by 12, 16,20 and 24.

Question2: a gardener with 5770 plants to arrange them such that the number of rows of plants equals the number of plants in each row. When he does so,he discovers that the number of plants falls short by 6 .find the number of rows the gardener wants to make.

1.

ok, you can forget worrying about the 12, as 24 is in the problem already (if it is divisible by 24, it is automatically divisble by 12).
consider
16=4x4
20=4x5
24=4x6
you have four fours, but you will need a 5 and 6 to make a perfect square
answer: 4^4 x 5^2 x 6^2=230400

Answer is 75

1=3600

1=3600

2=76

Question 1:

To find the least perfect squared number that is divisible by 12, 16, 20, and 24, we need to find the least common multiple (LCM) of these numbers.

Step 1: Prime factorize each number:
12 = 2^2 * 3
16 = 2^4
20 = 2^2 * 5
24 = 2^3 * 3

Step 2: Identify the highest power of each prime factor across all the numbers:
Highest power of 2 = 2^4
Highest power of 3 = 3
Highest power of 5 = 5

Step 3: Multiply the highest powers of the prime factors to calculate the LCM:
LCM = 2^4 * 3 * 5 = 80

The least perfect squared number that is divisible by 12, 16, 20, and 24 is 80.

Question 2:
Let's assume the number of rows the gardener wants to make is 'r'.

According to the given information, the number of plants in each row is also 'r', and the total number of plants he has is 5770.

We can set up the following equation based on the given information:

r * r = 5770 - 6

Simplifying the equation:

r^2 = 5764

To find the value of 'r', we need to find the square root of 5764.

Taking the square root of both sides:

r = √5764

Using a calculator, we find that the square root of 5764 is approximately 76. Therefore, the number of rows the gardener wants to make is 76.