自然数 1到25 的二次方，简直就是计算其他数字二次方的工具或原料，我们一起先熟悉一下儿吧。
一位数的二次方，我们都熟悉乘法口诀，我们就利用它们，先熟悉一下儿算式吧。5' = 5 X 5 = 25；6' = 6' - 4' + 4' = (6-4)(6+4) + 4' = 2 X 10 +16 = 36；7' = 7' - 3' + 3' = (7-3)(7+3) +3' = 4 X 10 + 9 = 49；8' = 8' - 2' + 2' = (8-2)(8+2) +2' = 6 X 10 + 4 = 64；9' = 9' - 1' + 1' = (9-1)(9+1) +1' = 8 X 10 + 1 = 81；这样看来，也难怪平方数的末位数就只有 0、1、4、9、6、5 这6个。
10' = 10 X 10 = 100；11' = (11+1)(11-1) + 1' = 12 X 10 + 1 = 121；12' = (12+2)(12-2) + 2' = 14 X 10 + 4 = 144；13' = (13+3)(13-3) + 3' = 16 X 10 + 9 = 169；14' = (14+4)(14-4) + 4' = 18 X 10 +16 = 196；
15' = 10 X 20 + 5' = 225；16' = (16-4)(16+4) + 4' = 12 X 20 +16 = 240+16 = 256；17' = (17-3)(17+3) + 3' = 14 X 20 + 9 = 280 + 9 = 289；18' = (18-2)(18+2) + 2' = 16 X 20 + 4 = 320 + 4 = 324；19' = (19-1)(19+1) + 1' = 18 X 20 + 1 = 360 + 1 = 361；
20' = 20 X 20 = 400；21' = (21+1)(21-1) + 1' = 22 X 20 + 1 = 440 + 1 = 441；22' = (22+2)(22-2) + 2' = 24 X 20 + 4 = 480 + 4 = 484；23' = (23+3)(23-3) + 3' = 26 X 20 + 9 = 520 + 9 = 529；24' = (24+4)(24-4) + 4' = 28 X 20 +16 = 560+16 = 576；
25' = 20 X 30 + 25 = 625；26' = (26-4)(26+4) + 4' = 22 X 30 +16 = 660+16 = 676；27' = (27-3)(27+3) + 3' = 24 X 30 + 9 = 720 + 9 = 729；28' = (28-2)(28+2) + 2' = 26 X 30 + 4 = 780 + 4 = 784；29' = (29-1)(29+1) + 1' = 28 X 30 + 1 = 840 + 1 = 841；看出来了吧，小于25 的数字，变成整十20 相乘还方便，可是大于25 的数字，变成整十30 乘起来进位就有些麻烦了。因此，简化计算光靠整十还远远不够，更多的数字我们还要变成 50，相乘得到整百来计算。
或者说，平方差这样就相当于完全平方式再做一个因式分解，前两项 a' 与 2ab 把公因式 a 提出来，就是平方差这样的效果了。 (a+b)' = a' +2ab +b' = a (a +2b) +b' = (a+b -b)(a+b +b) +b' (a-b)' = a' -2ab + b' = a (a -2b) + b' = (a-b +b)(a-b -b) + b' 两种算式对比看看，原来的二次方，完全平方的式子是 (a+b)' 或者(a-b)'，平方差的式子本来就是 a'；简化计算变成整十，平方差的式子要变成 (a+b)或者(a-b)，完全平方的整十本来就是 a
小于 30的自然数，二次方顶多只有三位数，其他数字计算二次方，就要用它们作 b' ，我们就要先熟悉它们。它们自己计算二次方，只要用 1到5 作 b，把 (a+b)或(a-b)变成整十，轻松相乘算出平方差，再把 b' 加回去。
两位数 25 到 50，b 就利用 1 到 25，把 (a+b) 变成 50。(a-b)就是 [(a+b)-2b]，这个 2b 肯定是偶数，(a+b)=50 也是偶数，那么(a-b) =[(a+b)-2b] 还是偶数，相乘得到的平方差就又是整百，这样再把 b' 加回去，整百与三位数相加也同样方便。
两位数 50 到 75，b 就利用 1 到 25，把 (a-b) 变成 50 来计算。(a+b)与(a-b) 相乘得到的平方差就还是整百，这样再把 b' 加回去，整百与三位数相加也同样方便。
两位数 75 到三位数 125，不用说相信大家也知道，b 利用 1 到 25，把 (a+b)或者(a-b) 变成 100 来计算，情况就与 25到75 的两位数差不多，算起来甚至感觉更简便。
500 前面的三位数，计算二次方，我们就根据最后的两位数，还是像这样计算。
方法分析到这里，我们来算一算 25 到 125 的二次方吧。好方法可要用一用才有感受啊！
25' = 20 X 30 + 25 = 625 26' = (26-24)(26+24) + 24' = 2 X 50 + 576 = 100 + 576 = 676 27' = (27-23)(27+23) + 23' = 4 X 50 + 529 = 200 + 529 = 729 28' = (28-22)(28+22) + 22' = 6 X 50 + 484 = 300 + 484 = 784 29' = (29-21)(29+21) + 21' = 8 X 50 + 441 = 400 + 441 = 841
30' = 30 X 30 = 900 31' = (31-19)(31+19) + 19' = 12 X 50 + 361 = 600 + 361 = 961 32' = (32-18)(32+18) + 18' = 14 X 50 + 324 = 700 + 324 = 1024 33' = (33-17)(33+17) + 17' = 16 X 50 + 289 = 800 + 289 = 1089 34' = (34-16)(34+16) + 16' = 18 X 50 + 256 = 900 + 256 = 1156
35' = 30 X 40 + 25 = 1225 36' = (36-14)(36+14) + 14' = 22 X 50 + 196 = 1100 + 196 = 1296 37' = (37-13)(37+13) + 13' = 24 X 50 + 169 = 1200 + 169 = 1369 38' = (38-12)(38+12) + 12' = 26 X 50 + 144 = 1300 + 144 = 1444 39' = (39-11)(39+11) + 11' = 28 X 50 + 121 = 1400 + 121 = 1521
40' = 40 X 40 = 1600 41' = (41-9)(41+9) + 9' = 32 X 50 + 81 = 1600 + 81 = 1681 42' = (42-8)(42+8) + 8' = 34 X 50 + 64 = 1700 + 64 = 1764 43' = (43-7)(43+7) + 7' = 36 X 50 + 49 = 1800 + 49 = 1849 44' = (44-6)(44+6) + 6' = 38 X 50 + 36 = 1900 + 36 = 1936
45' = 40 X 50 + 25 = 2025 46' = (46-4)(46+4) + 4' = 42 X 50 +16 = 2100 +16 = 2116 47' = (47-3)(47+3) + 3' = 44 X 50 + 9 = 2200 + 9 = 2209 48' = (48-2)(48+2) + 2' = 46 X 50 + 4 = 2300 + 4 = 2304 49' = (49-1)(49+1) + 1' = 48 X 50 + 1 = 2400 + 1 = 2401
50' = 50 X 50 = 2500 两位数 25到50 的二次方，我们全都算过了，(a+b)变成50，b 从1到25，我们都用过一遍了，后面其余数字的计算，(a-b)(a+b) 这一步我就省略不写了。 51' = 52 X 50 + 1' = 2600 + 1 = 2601 52' = 54 X 50 + 2' = 2700 + 4 = 2704 53' = 56 X 50 + 3' = 2800 + 9 = 2809 54' = 58 X 50 + 4' = 2900 +16 = 2916
55' = 60 X 50 + 25 = 3025 56' = 62 X 50 + 6' = 3100 + 36 = 3136 57' = 64 X 50 + 7' = 3200 + 49 = 3249 58' = 66 X 50 + 8' = 3300 + 64 = 3364 59' = 68 X 50 + 9' = 3400 + 81 = 3481
60' = 60 X 60 = 3600 61' = 72 X 50 + 11' = 3600 + 121 = 3721 62' = 74 X 50 + 12' = 3700 + 144 = 3844 63' = 76 X 50 + 13' = 3800 + 169 = 3969 64' = 78 X 50 + 14' = 3900 + 196 = 4096
65' = 60 X 70 + 25 = 4225 66' = 82 X 50 + 16' = 4100 + 256 = 4356 67' = 84 X 50 + 17' = 4200 + 289 = 4489 68' = 86 X 50 + 18' = 4300 + 324 = 4624 69' = 88 X 50 + 19' = 4400 + 361 = 4761
70' = 70 X 70 = 4900 71' = 92 X 50 + 21' = 4600 + 441 = 5041 72' = 94 X 50 + 22' = 4700 + 484 = 5184 73' = 96 X 50 + 23' = 4800 + 529 = 5329 74' = 98 X 50 + 24' = 4900 + 576 = 5476
75' = 70 X 80 + 25 = 5625 两位数 50到75 的二次方，我们也都算过了，(a-b)变成50，b 从1到25，我们又用过了一遍 76' = 52 X 100 + 24' = 5200 + 576 = 5776 77' = 54 X 100 + 23' = 5400 + 529 = 5929 78' = 56 X 100 + 24' = 5600 + 484 = 6084 79' = 58 X 100 + 21' = 5800 + 441 = 6241
80' = 80 X 80 = 6400 81' = 62 X 100 + 19' = 6200 + 361 = 6561 82' = 64 X 100 + 18' = 6400 + 324 = 6724 83' = 66 X 100 + 17' = 6600 + 289 = 6889 84' = 68 X 100 + 16' = 6800 + 256 = 7056
85' = 80 X 90 + 25 = 7225 86' = 72 X 100 + 14' = 7200 + 196 = 7396 87' = 74 X 100 + 13' = 7400 + 169 = 7569 88' = 76 X 100 + 12' = 7600 + 144 = 7744 89' = 78 X 100 + 11' = 7800 + 121 = 7921
90' = 90 X 90 = 8100 91' = 82 X 100 + 9' = 8200 + 81 = 8281 92' = 84 X 100 + 8' = 8400 + 64 = 8464 93' = 86 X 100 + 7' = 8600 + 49 = 8649 94' = 88 X 100 + 6' = 8800 + 36 = 8836
95' = 90 X 100 + 25 = 9025 96' = 92 X 100 + 4' = 9200 +16 = 9216 97' = 94 X 100 + 3' = 9400 + 9 = 9409 98' = 96 X 100 + 2' = 9600 + 4 = 9604 99' = 98 X 100 + 1' = 9800 + 1 = 9801
100' = 100 X 100 = 10000 前面的 75到100 算平方，就要用 b 从 1到25，把 (a+b)变成100， 接下来 100到125 算平方，又是用 b 从 1到25，把 (a-b)变成100 来计算。 101' = 102 X 100 + 1' = 10200 + 1 = 10201 102' = 104 X 100 + 2' = 10400 + 4 = 10404 103' = 106 X 100 + 3' = 10600 + 9 = 10609 104' = 108 X 100 + 4' = 10800 + 16 = 10816
105' = 110 X 100 + 5' = 11000 + 25 = 11025 106' = 112 X 100 + 6' = 11200 + 36 = 11236 107' = 114 X 100 + 7' = 11400 + 49 = 11449 108' = 116 X 100 + 8' = 11600 + 64 = 11664 109' = 118 X 100 + 9' = 11800 + 81 = 11881
110' = 110 X 110 = 12100 111' = 122 X 100 + 11' = 12200 + 121 = 12321 112' = 124 X 100 + 12' = 12400 + 144 = 12544 113' = 126 X 100 + 13' = 12600 + 169 = 12769 114' = 128 X 100 + 14' = 12800 + 196 = 12996
115' = 130 X 100 + 15' = 13000 + 225 = 13225 116' = 132 X 100 + 16' = 13200 + 256 = 13456 117' = 134 X 100 + 17' = 13400 + 289 = 13689 118' = 136 X 100 + 18' = 13600 + 324 = 13924 119' = 138 X 100 + 19' = 13800 + 361 = 14161
120' = 120 X 120 = 14400 121' = 142 X 100 + 21' = 14200 + 441 = 14641 122' = 144 X 100 + 22' = 14400 + 484 = 14884 123' = 146 X 100 + 23' = 14600 + 529 = 15129 124' = 148 X 100 + 24' = 14800 + 576 = 15376
125' = 150 X 100 + 25' = 15000 + 625 = 15625 数字这样计算二次方，感觉是不是方便多了呢？
正如平方数的末位数就只有 0、1、4、9、6、5 这6个，平方数的最后两位，也只有 1到25 这二十几个，有趣吧。十位数是 0、4、5、9，二次方的后两位，也就是 1到9 的平方值十位数是 1、3、6、8，二次方的后两位就都是 21、44、69、96、25、56、89、24、61十位数是 2、7，二次方的后两位也都是 41、84、29、76、25 END
连续的二次方计算了这么多，我们再算一些不连续的数字，就算算这些三位数的平方数吧。
100' = 100 X 100 = 10000
121' = 100 X (121+21) +21' = 142 X 100 +441 = 14641
144' = 150 X (144 - 6) + 6' = 138 X 150 +36 = 69 X 300 +36 = 20736
169' = 150 X (169+19) +19' = 188 X 150 +361 = 94 X 300 +361 = 28561
196' = 200 X (196 - 4) + 4' = 192 X 200 +16 = 38400 +16 = 38416
225' = 200 X 250 +625 = 50625
256' = 250 X (256+6) +6' = 262 X 1000 / 4 +36 = 65500 +36 = 65536
289' = 300 X (289-11) +11' = 278 X 300 +121 = 83400 +121 = 83521
324' = 300 X (324+24) +24' = 348 X 300 +576 = 104400 +576 = 104976
361' = 350 X (361+11) +11' = 372 X 350 +121 = 186 X 700 +121 = 130321
400' = 400 X 400 = 160000
441' = 450 X (441-9) +9' = 432 X 450 +81 = 216 X 900 +81 = 194481
484' = 500 X (484-16) +16' = 468 X 500 +256 = 234000 +256 = 234256
529' = 550 X (529-21) +21' = 508 X 550 +441 = 279400 +441 = 279841
576' = 600 X (576-24) +24' = 552 X 600 +576 = 331200 +576 = 331776
625' = 600 X 650 +625 = 390625
676' = 700 X (676-24) +24' = 652 X 700 +576 = 456400 +576 = 456976
729' = 750 X (729-21) +21' = 708 X 750 +441 = 2124000 / 4 +441 = 531441
784' = 800 X (784-16) +16' = 768 X 800 +256 = 614400 +256 = 614656
841' = 800 X (841+41) +41' = 882 X 800 +1681 = 705600 +1681 =707281
900' = 900 X 900 = 810000
961' = 1000 X (961-39) +39' = 922000 +1521 = 923521
怎么样，看出来了吗？为什么我说任意的一个数字必须选自 1到500 之间呢？就是因为小于500 的数字变成50，除了 250 还可以用 1000 / 4 算得方便，其余的 150、350、450，只要乘以2 就是 300、700、900，相乘同样算得简便。
可是大于500 的数字变成50，除了 750 可用 3000 / 4 算得方便，其余 550、650、850、950，乘以2 还是 1100、1300、1700、1900，这样乘起来还是不方便啊。