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平方数の下二桁について

数	平方数	平方数	数
1	1	2401	49
2	4	2304	48
3	9	2209	47
4	16	2116	46
5	25	2025	45
6	36	1936	44
7	49	1849	43
8	64	1764	42
9	81	1681	41
10	100	1600	40
11	121	1521	39
12	144	1444	38
13	169	1369	37
14	196	1296	36
15	225	1225	35
16	256	1156	34
17	289	1089	33
18	324	1024	32
19	361	961	31
20	400	900	30
21	441	841	29
22	484	784	28
23	529	729	27
24	576	676	26
25	625	625	25

$24 \times 24 = 576$ と $26 \times 26 = 676$ の下二桁は同じです。同様に $23 \times 23 = 529$ と $27 \times 27 = 729$ も下二桁が一致しています。この性質は

$d = (25 + a) (25 + a) - (25 - a) (25 - a)$

が $100$ の倍数になることから証明できます。

$d = (25 + a) (25 + a) - (25 - a) (25 - a) = 625 + 50 a + a^{2} - (625 - 50 a + a^{2}) = 100 a$

下二桁の性質を理解すると平方数を覚えやすくなります。

平方数の一覧と平方数の下二桁の性質。例えば 23×23 の 529 と 27×27 の729 は下二桁が 29 と一致しています。