IB MATHEMATICS Topic : Sequences and Series – Patterns in Powers of Numbers

Consider all the powers from 1 to 10 of the numbers one to 10.

1	1	1	1	1	1	1	1	1	1	1
2	2	4	8	16	32	64	128	256	512	1024
3	3	9	27	81	243	729	2187	6561	19683	59049
4	4	16	64	256	1024	4096	16384	65536	262144	0148576
5	5	25	125	625	3125	15625	78125	390625	1953125	9765625
6	6	36	216	1296	7776	46656	279936	16796160	10077696	60466176
7	7	49	343	2401	16807	117649	823543	5764801	40353607	282475249
8	8	64	512	4096	32768	262144	2097152	16777216	134217728	1073741824
9	9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401
10	10	100	1000	10000	100000	1000000	10000000	100000000	1000000000	10000000000

We are only interested on the last digit of each entry in the table above. These are shown in the table below.

1	1	1	1	1	1	1	1	1	1	1
2	2	4	8	6	2	4	8	6	2	4
3	3	9	7	1	3	9	7	1	3	9
4	4	6	4	6	4	6	4	6	4	6
5	5	5	5	5	5	5	5	5	25	25
6	6	6	6	6	6	6	6	6	6	6
7	7	9	3	1	7	9	3	1	7	9
8	8	4	2	6	8	4	2	6	8	4
9	9	1	9	1	9	1	9	1	9	1
10	10	100	1000	10000	100000	1000000	10000000	100000000	1000000000	10000000000

Notice that the last digits ofandare constant – they are 1 and 6 respectively for alland that 6-1=5.

Notice that the last digits ofandmake cycles of length 4: they are 2,4,8,6 and 7,9,3,1 respectively, and that 7-2 =5.

Notice that the last digits ofandmake cycles of length 4: they are 3,9,7,1 and 8,4,2,6 respectively, and that 8-3 =5.

Notice that the last digits ofandmake cycles of length 2: they are 4,6 and 9,1 respectively, and that 9-4 =5.

These patterns exist because ifthenso the last digits are related by a difference of 5 at each stage. If one oforhas a cycle of a certain length, so must the other.