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Finding the Square of a Number

We can easily find the square of one-digit and smaller two-digit numbers. But when the number
becomes larger, it becomes laborious and time consuming. Let us see some methods to find the
square of numbers quickly, without doing the actual multiplication.
Let us find square of 46. Quick Check

We know that 46 = 40 + 6 Find the square of:
2
2
Therefore, 46 = (40 + 6) = (40 + 6) × (40 + 6) 1. 58 2. 73
2
= 40(40 + 6) + 6(40 + 6) = 40 + 40 × 6 + 6 × 40 + 6 2
= 1600 + 240 + 240 + 36 = 2116

Square of Natural Numbers Ending with 5

Observe the following:

2
25 = 625 = (2 × 3) hundreds + 25
2
35 = 1225 = (3 × 4) hundreds + 25
2
125 = 15625 = (12 × 13) hundreds + 25
Thus, we can find the square of numbers with unit digit 5, easily as follows:

First we cross out 5 and multiply the remaining numbers by its successor, and at last suffix 25 to
the product to obtain the square.
2
Moreover, (a5) = (10a + 5)² = (10a + 5) (10a + 5)
= 10a × 10a + 10a × 5 + 5 × 10a + 5 × 5

= 100a +100a + 25 = 100a (a + 1) + 25
2
= a(a + 1) hundreds + 25

Example 17: Find the square of the following:
(a) 95 (b) 105 (c) 525

2
Solution: (a) 95 = 9 (9 + 1) hundreds + 25 = (9 × 10) hundreds + 25 = 9000 + 25 = 9025
2
(b) 105 = 10(10 + 1) hundreds + 25 = (10 × 11) hundreds + 25 = 11025
2
(c) 525 = 52(52 + 1) hundreds + 25 = (52 × 53) hundreds + 25 = 275600 + 25 = 275625
Pythagorean Triplets

In earlier class, we learnt that 3 cm, 4 cm and 5 cm are the side lengths of a right-
angled triangle because
5 4
3 + 4 = 9 + 16 = 25 = 5 2
2
2
The set of three numbers such as 3, 4 and 5 is known as Pythagorean triplet. A set
2
2
of three numbers a, b and c is called a Pythagorean Triplet if a + b = c . 3
2
Some other examples of Pythagorean triplets are 5, 12, 13 and 9, 12, 15.
2
2
2
2
2
For any natural number m > 1, (2m) + (m – 1) = (m + 1) . So, 2m, (m – 1) and (m + 1) form a
2
2
Pythagorean triplet.
133 Squares and Square Roots