Find the least number that should be added to 1266 to make it a perfect square. Also, find the square root of the perfect square.
Find the greatest number of 5 digits which is a perfect square. Also, find its square root.
Answer (2)
To make 1266 a perfect square, add 30 to get 1296, whose square root is 36. The greatest 5-digit perfect square is 99856, with a square root of 316.
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To solve the two problems given:
Finding the Least Number to be Added to 1266 to Make It a Perfect Square and Its Square Root:
First, identify the integer that, when squared, is close to 1266 but not greater than it:
n 2 ≤ 1266 < ( n + 1 ) 2
Using trial and error or calculation, we find:
3 5 2 = 1225 3 6 2 = 1296
Since 1266 is between 35² and 36², it's closer to 36², which means we add a number to reach 36².
3 6 2 − 1266 = 1296 − 1266 = 30
Therefore, the least number to be added to 1266 to make it a perfect square is 30. The square root of the resulting perfect square, 1296, is 36.
Finding the Greatest Number of 5 Digits Which is a Perfect Square and Its Square Root:
The greatest 5-digit number is 99999. We need to find the largest perfect square less than or equal to this number.
Start by estimating the square root of 99999:
99999 ≈ 316.227 (using a calculator)
The perfect square will be:
31 6 2 = 99856
Since 317² is greater than 99999, the greatest perfect square less than 100000 is 99856.
Therefore, the greatest 5-digit number that is a perfect square is 99856, and its square root is 316.
