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