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Finding the Square of a Number - Other Patterns in Squares

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Find the square of 23 so quickly . The answer is not so easy and we may need to multiply 23 by 23. There is a way to find this without having to multiply 23 × 23. 
We know 23 = 20 + 3
 Therefore  `23^2` 
= `(20 + 3)^2` = 20(20 + 3) + 3(20 + 3) 
= `20^2 + 20 × 3 + 3 × 20 + 3^2` 
= 400 + 60 + 60 + 9 = 529 

1)  Other patterns in squares :
Consider a number with unit digit 5, i.e., a5 
`(a5)^2 = (10a + 5)^2` 
= 10a(10a + 5) + 5(10a + 5) 
= `100a^2 + 50a + 50a + 25 `
= 100a(a + 1) + 25 
= a(a + 1) hundred + 25 
Consider the following pattern: 
252 = 625 = (2 × 3) hundreds + 25 
352 = 1225 = (3 × 4) hundreds + 25 

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