A number that ends in 5 has the form A = 10a + 5, where a has one digit less than A. To find the square A² of A, append 25 to the product a×(a + 1) of a with its successor. For example, compute 115². 115 = 11·10 + 5, so that a = 11. First compute 11·(11 + 1) = 11·12 = 132 (since 3 = 1 + 2). Next, append 25 to the right of 132 to get 13225!
Why does this work?
A²
= (10a + 5)²
= 100a² + 2·10a·5 + 25
= 100a(a + 1) + 25.
Which exactly means that 25 is appended to the product a(a + 1).
Another example: to compute 245² observe that a = 24. Then
24·(24 + 1) = 24² + 24 = 576 + 24 = 600.
Therefore 245²=60025. Here is another way to compute 24·25:
24·25 = 2400/4 = 1200/2 = 600.
The rule naturally applies to 2-digit numbers as well. 75² = 5625 (since 7·8 = 56).
