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Home >> Numbers >> Square Numbers >> Properties of Square Numbers >> Property 2 >>

If Natural Number has 1 or 9 at one's place, then Square of such Natural Numbers must have 1 at one's place

Property 1 Property 2 Property 3 Property 4 Property 5
Property 6 Property 7 Property 8

Before you understand this property, you are adviced to read:

What are Natural Numbers ?
What are Square Numbers ?

Observe the following Table 1:

Table 1

Natural Number Square Number
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400


Above table - 1 represents square of first 20 natural numbers.
You must have observed that the Natural Numbers having 1 or 9 at its one's place (highlighted in yellow), their corresponding Square Numbers have 1 at its one's place (highlighted in green).
i.e. Square of 1 = 1
Square of 9 = 81
Square of 11 = 121
Square of 19 = 361

Let's try few more square numbers as shown in table 2 and table 3:

Table : 2 Table : 3
Natural Number Square Number
41 1681
42 1764
43 1849
44 1936
45 2025
46 2116
47 2209
48 2304
49 2401
50 2500
51 2601
52 2704
53 2809
54 2916
55 3025
56 3136
57 3249
58 3364
59 3481
60 3600
61 3721
Natural Number Square Number
71 5041
72 5184
73 5329
74 5476
75 5625
76 5776
77 5929
78 6084
79 6241
80 6400
81 6561
82 6724
83 6889
84 7056
85 7225
86 7396
87 7569
88 7744
89 7921
90 8100
91 8281


Table - 2 represents squares of numbers from 41 to 61
Table - 3 represents squares of numbers from 71 to 91

In these tables also you will observe the same pattern which you observed in Table 1 i.e. Natural Numbers having 1 or 9 at its one's place (highlighted in yellow), their corresponding Square Numbers have 1 at its one's place (highlighted in green).

So, this explains this property of square numbers that:
If a Natural Number has 1 or 9 at its one's place, then the Square of such Natural Numbers must have 1 at its one's place.

Also we get that:
If a Natural Number has 1 at its one's place, then the Natural Number is not always a Perfect Square.
e.g. 41, 71, 111 all are not square numbers

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