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Properties of Square Numbers

The following table contains squares of numbers from 1 to 100.

Number Square Number Square Number Square Number Square
1 1 26 676 51 2601 76 5776

2 4 27 729 52 2704 77 5929
3 9 28 784 53 2809 78 6084
4 16 29 841 54 2916 79 6241
5 25 30 900 55 3025 80 6400
6 36 31 961 56 3136 81 6561
7 49 32 1024 57 3249 82 6724
8 64 33 1089 58 3364 83 6889
9 81 34 1156 59 3481 84 7056

10 100 35 1225 60 3600 85 7225
11 121 36 1296 61 3721 86 7396
12 144 37 1369 62 3844 87 7569
13 169 38 1444 63 3969 88 7744
14 196 39 1521 64 4096 89 7921
15 225 40 1600 65 4225 90 8100

16 256 41 1681 66 4356 91 8281
17 289 42 1764 67 4489 92 8464
18 324 43 1849 68 4624 93 8649
19 361 44 1936 69 4761 94 8836
20 400 45 2025 70 4900 95 9025
21 441 46 2116 71 5041 96 9216
22 484 47 2209 72 5184 97 9409

23 529 48 2304 73 5329 98 9604
24 576 49 2401 74 5476 99 9801
25 625 50 2500 75 5625 100 10,000
Let us understand the properties of square numbers.

Property 1

From the above table, we can observe that the unit digits (i.e., digit in the ones place) of square
numbers are 0, 1, 4, 5, 6 or 9. None of these end with 2, 3, 7 or 8 at the units place.
But we cannot say that if a number ends in 0, 1, 4, 5, 6 or 9 is always a perfect square number.
For example, 50, 51, 34, 45, 46, 39, etc. are not perfect squares. Thus, we can say that

“A square number always ends in 0, 1, 4, 5, 6 or 9” and the number having 2, 3, 7 or 8 at unit place
is never a square number.

Example 7: Can we say whether the following numbers are perfect squares? Justify your answer.
(a) 7777 (b) 968

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