find least number to be subtracted from 1886 to get a perfect square

Question

 for .the given number ,find the leat number to be subtracted to get a perfect square. the number is 1886

Answer 1

√1886=43.428102...

43^2=1849

1886-1849=37 is to be subtracted

Answer 2

=Method 1.  List the perfect squares less than 1886, find the one just less than 1886, then subtract.

 

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

 

1886 - 1849 = 37

 

Method 2.  Estimate the square root of 1886 first.

  √(1886) = √(100*18.86) = 10√18.86

  So, the square root is between 10*4 and 10*5  (that is, 40 and 50).

  40²=1600, 50²=2500, and 45² = 2025

  Now, split 40²=1600 and 45² = 2025:

     40² = 1600, 43² = 1849

  The perfect square must be between 1849 (43²) and 2025 (45²).

  Now, split 43²=1849 and 45²=2025:

     44⁴ = 1936   [that's too big]

 

The perfect square must be 43²=1849.  Subtract that from 1886.

 

{Note: before calculators, we learned how to do the Newton-Raphson method using slide rules.)

Answer 3

We know that 4² = 16, and 40² = 1600

 

Let's look at numbers after 40:

 

41² = 1681

 

42² = 1764

 

43² = 1849

 

44² = 1936

 

Since 1936 > 1886, we can stop at 44. The largest perfect square before the number 1886 is 1849 = 43²

 

If we subtract 1886 - 1849, we get 37. This is the least number to be subtracted from 1886 to obtain a perfect square.