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

Powers Table

A. Describe patterns you see in the ones digits of the powers.

B. Predict the ones digit in the following:

1. 4¹²

2. 9²⁰

3. 3¹⁷

4. 5¹⁰⁰

5. 10⁵⁰⁰

C. Predict the ones digit in the following:

1. 31¹⁰

2. 12¹⁰

3. 17²¹

4. 29¹⁰

A. For numbers 2,3,7 and 8, pattern is of 4 digits.

For numbers 4 and 9, pattern is of 2 digits.

For numbers 1,5,6 and 10, pattern is of 1 digit.

B. 1. Pattern in number 4 is of 2 digits, 4 and 6.

On dividing power of 4, that is 12, by 2 and checking the remainder , we can predict the one’s digit in 4¹²

⇒

⇒ 4⁰ (on diving 12 by 2 we get remainder 0)

one’s digit is = 6 (second number out of 4 and 6)

2. Pattern in number 9 is of 2 digits, 9 and 1.

On dividing power of 9, that is 20, by 2 and checking the remainder , we can predict the one’s digit in 9²⁰

⇒

⇒ 9¹⁰ (on dividing 20 by 2 we get remainder 0)

one’s digit is = 1 (second number out of 9 and 1)

3. Pattern in number 3 is of 4 digits, 3,9,7 and 1.

On dividing power of 3, that is 17, by 4 and checking the remainder , we can predict the one’s digit in 3¹⁷

⇒

⇒ 3¹ (on dividing 17 by 4 we get remainder 1)

one’s digit is = 3 (first number out of 3,9,7 and 1)

4. Pattern in number 5 is of 1 digit, 5.

one’s digit is = 5

5. Pattern in number 10 is of 1 digit, 0.

one’s digit is = 0

C. 1. Pattern in number 1 is of 1 digit, 1.

one’s digit is = 1

2. Pattern in number 2 is of 4 digits, 2,4,8 and 6.

On dividing power of 12, that is 10, by 4 and checking the remainder , we can predict the one’s digit in 12¹⁰

⇒

⇒ 12² (on dividing 10 by 4 we get remainder 2)

one’s digit is = 4 (second number out of 2,4,8 and 6)

3. Pattern in number 7 is of 4 digits,7,9,3 and 1.

On dividing power of 17, that is 21, by 4 and checking the remainder , we can predict the one’s digit in 17²¹

⇒

⇒ 17¹ (on dividing 21 by 4 we get remainder 1)

one’s digit is = 7 (first number out of 7,9,3 and 1)

4. Pattern in number 9 is of 2 digits,9 and 1.

On dividing power of 29, that is 10, by 2 and checking the remainder , we can predict the one’s digit in 29¹⁰

⇒

⇒ 29⁰ (on dividing 10 by 2 we get remainder 0)

one’s digit is = 1 (last number out of 9 and 1)