Study Notes for CAT 2021: Cyclicity

Note: This article is an original contribution of Tushar Dhingra, II MBA at IIT Delhi

Basic Concept of Cyclicity

The concept of cyclicity is used to identify the last digit of the number.

Let’s take an example to understand this:

Example:  Find the unit digit of 3⁵⁴.

Solution:  Now it’s a very big term and not easy to calculate but we can find the last digit by using the concept of cyclicity. We observe powers of 3

3¹ = 3

3² = 9

3³ = 27

3⁴ = 81

3⁵ = 243

3⁶= 729

So now pay attention to the last digits we can observe that the last digit repeats itself after a cycle of 4 and the cycle is 3, 9, 7, 1 this repetition of numbers after a particular stage is called the cyclicity of numbers.

Therefore, when we need to find the unit digit of any number like 3ⁿ we just need to find the number on which the cycle halts. So we divide power n by 4 to check remainder

• If remainder is 1 then the unit digit will be 3
• If remainder is 2 then the unit digit will be 9
• If remainder is 3 then the unit digit will be 7
• If remainder is 0 then the unit digit will be 1

We divided the power by 4 because cycle repeats itself after 4 values. Now the main question was that how much is the last digit of 3⁵⁴ and we know the cycle repeat itself after 4 so we will divide the 54 with 4, so on dividing 54 by 4 the remainder becomes 2.

Now as we discussed above if the remainder is 2 the last digit would be 9. Hence unit digit of 3⁵⁴ is 9.

Example: What will be the unit digit of 347⁴⁵

Solution: Let’s observe unit digit of 347 x 347 = 9.

The main purpose of the above is that the unit digit of any multiplication depends upon the unit digit of numbers, whatever is the number big or small the unit digit always depends upon the multiplication of the last digit.

So the last digit of 347⁴⁵ can be found by calculating last digit of 7⁴⁵

We observe unit digit while calculating powers of 7

7¹ = 7

7² = 49

7³ = 343

7⁴ = 2401

7⁵ = 16807

So on dividing 45 with 4, 1 will be the remainder and the last digit would be 7

Application of Concept of Cyclicity

How to calculate unit digit if a number contains power of a power

Example: What will be the last digit of (12²³)⁴⁵

Solution:

To find the last digit of this type of number we will start the question from the base the base is given to be 12.

It means we will see the cyclicity of 2 because the last digit depends upon the unit digit of 12 i.e. 2.

We observe unit digit while calculating powers of 2

2¹ = 2

2² = 4

2³ = 8

2⁴ = 6

2⁵ = 2

Step 1: Now we know that cyclicity of last digit of 12 i.e. 2 is of 4, hence we divide the power of 12 i.e. 23⁴⁵ with 4.

Step 2: Now let’s calculate the remainder of 23⁴⁵ when divided by 4 and then we will determine the last digit.

Step 3:  The remainder will be 3 because we can write remainder of 23 by 4 as 3 or -1. Hence we can write 23⁴⁵ as (-1)⁴⁵ but odd power of -1 will be again -1 and thus 23⁴⁵ when divided by 4 will give us remainder as -1 or 3.

Hence unit digit of (12²³ )⁴⁵will be same as unit digit of 2³ i.e. 8

Example: Find the unit digit of (32²⁵)⁹⁵? 

Solution: To find the last digit of this type of number we will start the question from the base the base is given to be 32. It means we will see the cyclicity of 2 because the last digit depends upon the unit digit of 32 i.e. 2.

We observe unit digit while calculating powers of 2

2¹ = 2

2² = 4

2³ = 8

2⁴ = 6

2⁵ = 2

Step 1: Now we know that cyclicity of last digit of 32 i.e. 2 is of 4, hence the divide the power of 12 i.e. 25⁹⁵ with 4.

Step 2: Now let’s calculate the remainder of 25⁹⁵ when divided by 4 and then we will determine the last digit.

Step 3:  The remainder will be 1 because we can write remainder of 25 by 4 as 1. Hence we can write 2595 as (1) 95 but any power of 1 will be again 1 and thus 25⁹⁵ when divided by 4 will give us remainder as 1.

Hence unit digit of (32²⁵)⁹⁵   will be same as unit digit of 2¹ i.e. 2

Questions: Find the unit digit of 4⁶⁸⁶

1. a) 4 b) 6 c) 2 d) 3

Answer: b

Solution: We know

4¹ = 4

4² = 6

4³ = 4

4⁴ = 6

Here, we see that powers of four repeat after a cycle of 2 i.e.

Odd power of 4 gives unit digit as 4 and even power of 4 gives unit digit as 6

So unit digit of 4⁶⁸⁶ is 6.

Question: What will be the unit digit of 65¹²³⁴

1. a) 5 b) 0  b)  2  d) 3

Answer: a

Solution: We know

5¹ = 5

5² = 25

5³ = 125

5⁴ = 625

So, we observe that unit digit of any power of 5 is 5. Hence unit digit of 65¹²³⁴ is 5.

Question:  Find the unit digit of 36⁵¹²

1. a) 4 b) 6 c) 2 d) 3

Answer: b

Solution: We know

6¹ = 6

6² = 36

6³ = 216

So, we observe that unit digit of any power of 6 is 6. Hence unit digit of 36⁵¹² is 6.

Question: Find the unit digit of 18⁵⁶⁹³

1. a) 4 b) 6 c) 2 d) 8

Answer: d

Solution: We observe unit digit while calculating powers of 8

Unit digit in 8¹ = 8

Unit digit in 8² = 4

Unit digit in 8³ = 2

Unit digit in 8⁴ = 6

Unit digit in 8⁵ = 8

So on dividing 5693 with 4, 1 will be the remainder and hence the last digit would be 8.

Universal Cyclicity: Since all the numbers start repeating themselves after 4, the universal cyclicity of all numbers is 4.

Example: Calculate 578¹⁰²⁹

Solution: Let`s look at the following steps.

Step 1: We divide the exponent of the number by 4. So we divide 1029 by 4 and get the remainder as 1. Step 2: Now the unit`s digit of the number can be calculated by solving 8<sup>1</sup>. Here 8 is the unit`s digit of the number 578 and 1 is the remainder.

Step 3: If the remainder is zero, then the unit digit will be same as the unit digit of N⁴.

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