Squaring

If last n digits are n 0s: Square the number to the left of 0s and double the number of 0s. For example, 4500^2 = 20250000.

If last digit is 1: Say the number is 41. Square of 41 = 40^2 + 40 + 41 = 1681.

If last digit is 2: Say the number is 152. Square of 152 = 150^2 +2*(150 x 152) = 23104.

If the last digit is 3: Say the number is 153. 153^2 = 150^2 + 3*(153+150) = 23409. 

If the last digit is 4: Say the number is 154. 154^2 = 150^2 + 4*(154+150) = 23716. Or, 154^2 = 155^2 – (155+154) = 23176.

If last digit is 5: Say the number is 45. Take 4. Multiply (4+1). Square it. Square of 45 = 4x (4+1) 25 = 2025. Take another number – 4785. Square of 785 = 78x79 25 = 608425.

If the last digit is 6: Say the number is 156. 156^2 = 160^2 - 4*(156+160) = 24336. Or, 156^2 = 155^2 + (155+156) = 24336.

If the last digit is 7: Say the number is 157. 153^2 = 160^2 - 3*(157+160) = 23409.

If the last digit is 8: Say the number is 158. 158^2 = 160^2 - 2*(158+160) = 24964.  Or, 158^2 = 155^2 + 3*(155+158) = 24964.

If the last digit is 9: Say the number is 159. 159^2 = 160^2 - (159+160) = 25281.

Thus, we can generalize the square of a number as –

if a>b, a^2 = b^2 + (a-b)*(a+b),

 
  or, if a<b, a^2 = b^2 – (b-a)*(b+a).