0a) Consider (ax + b)2 X a2. x2 + 2abx + b2.
This identity for x = 10 and b = 5 becomes
(10a + 5) 2 = a2 . 102 + 2. 10a . 5 + 52
= a2 . 102 + a. 102 + 52
= (a 2+ a ) . 102 + 52
= a (a + 1) . 10 2 + 25.
Clearly 10a + 5 represents two-digit numbers 15, 25, 35, -------,95 for the
values a = 1, 2, 3, -------,9 respectively. In such a case the number (10a + 5)2 is
of the form whose L.H.S is a (a + 1) and R.H.S is 25, that is, a (a + 1) / 25.
Thus any such two digit number gives the result in the same fashion.
