Question

Predicting the ones digit, copy and complete this table and answer the questions that follow.

Powers Table
x	1x	2x	3x	4x	5x	6x	7x	8x	9x	10x
1	1	2
2	1	4
3	1	8
4	1	16
5	1	32
6	1	64
7	1	128
8	1	256
Ones Digits of the Powers	1	2, 4, 8, 6

1. Describe patterns you see in the ones digits of the powers.
2. Predict the ones digit in the following:
   1. 4¹²
   2. 9²⁰
   3. 3¹⁷
   4. 5¹⁰⁰
   5. 10⁵⁰⁰
3. Predict the ones digit in the following:
   1. 31¹⁰
   2. 12¹⁰
   3. 17²¹
   4. 29¹⁰ 

Answer 1

a. On the basis of given pattern in 1^x and 2^x, we can make more patterns for 3^x, 4^x, 5^x, 6^x, 7^x, 8^x, 9^x, 10^x.

Thus, we have following table which shows all details about the patterns.

Powers Table
x	1x	2x	3x	4x	5x	6x	7x	8x	9x	10x
1	1	2	3	4	5	6	7	8	9	10
2	1	4	9	16	25	36	49	64	81	100
3	1	8	27	64	125	216	343	512	729	1000
4	1	16	81	256	625	1296	2401	4096	6561	10000
5	1	32	243	1024	3125	7776	16807	32768	59049	100000
6	1	64	729	4096	15625	46656	117649	262144	531441	1000000
7	1	128	2187	16384	78125	279936	823543	2097152	4782969	10000000
8	1	256	6561	65536	390625	1679616	5764801	16777216	43046721	100000000
Ones Digits of the Powers	1	2, 4, 8, 6	3, 9, 7, 1	4, 6	5	6	7, 9, 3, 1	8, 4, 2, 6	9.1	0

b. 

1. Ones digit in 4¹² is 6.
2. Ones digit in 9²⁰ is 1.
3. Ones digit in 3¹⁷ is 3.
4. Ones digit in 5¹⁰⁰ is 5.
5. Ones digit in 10⁵⁰⁰ is 0.

Note: Follow the above given table.

c.

1. Ones digit in 31¹⁰ is 1.
2. Ones digit in 12¹⁰ is 4.
3. Ones digit in 17²¹ is 7.
4. Ones digit in 29¹⁰ is 1.