PATTERNS

Exponent (mod k) : 

                                                                   Exponent (mod k) means the remainder when exponent is divided by k.









PATTERNS OF 2

Cycle in the Last Digit of 2n, n≥1

Power of Two (k ≥ 0)
Exponent (mod 4)
Last Digit
21+4k
1
2
22+4k
2
4
23+4k
3
8
24+4k
0
6



PATTERNS OF 3
Cycle in the Last Digit of 3n, n≥1

Power of Three (k ≥ 0)
Exponent (mod 4)
Last Digit
31+4k
1
3
32+4k
2
9
33+4k
3
7
34+4k
0
1



PATTERNS OF 7
Cycle in the Last Digit of 7n, n≥1

Power of seven (k ≥ 0)
Exponent (mod 4)
Last Digit
71+4k
1
7
72+4k
2
9
73+4k
3
3
74+4k
0
1






PATTERNS OF 8


Cycle in the Last Digit of 8n, n≥1

Power of Eight (k ≥ 0)
Exponent (mod 4)
Last Digit
81+4k
1
8
82+4k
2
4
83+4k
3
2
84+4k
0
6






PATTERNS OF 4 &9:

  power patterns of 4,9 repeats for every two numbers

     the number(4 power odd) 41+2k  ends with 4

     the number (4 power even) 42k   ends with 6

     the number( 9 power odd) 91+2k ends with 9

     the number (9 power even) 92k    ends with 1

PATTERNS OF 1,5 &9: 

the number(1 power positive integer) 1k  ends with 1

 the number(5 power positive integer) 5k  ends with 5

 the number(6 power positive integer) 6k  ends with 6