  Cyclicity of Remainder

In this section we learn about the cyclicity of  remainder. And  explain the concept of cyclicity of remainder with the help of suitable examples. For cat aspirants want to know more visit here

Concepts for CAT

In CAT, MAT and other Competitive examinations like Bank PO, etc. you get questions where you need to find the last digit of numbers raised to large powers. It’s almost impossible to calculate the values of such numbers manually and  hence to find digit at their unit’s place. Such problems can be solved using the concept commonly known as Cyclicity of Numbers.

Number system is the most fundamental topic in mathematics. It is a vast topic which mainly includes concept of HCF and LCM, concept of unit digit, concept of factors, concept of cyclicity, concept of factorials, etc.

Cyclicity of remainders is an important concept which can be used to solve questions based on remainders. This concept utilizes the fact that remainders repeat themselves after a certain interval when divided by a number.

Remainders come in a cycle.

When dividing by 7: 8 has remainder 1, 9 has remainder 2, 10 has remainder 3, 11 has remainder 4, 12 has remainder 5, 13 has remainder 6, 14 has remainder 0, 15 has remainder 1, 16 has remainder 2, 17 has remainder 3… etc. etc. etc., onward and forever. That is, when dividing by 7, the remainders cycle from 0-6 continuously. This is a general rule:

When dividing integers by integer n, there are n possible remainders, 0 through (n-1), and these remainders cycle in an infinite loop.I call this the ‘remainder cycle.’

Note that you can say something like “2 ÷ 9 is 0 remainder 2,” so it actually is said to have a remainder 2.

First of all, we know that Remainder = 0 to d – 1; where d= number by which the divisor is divided.

If we divide an by d, the remainder can be any value from 0 to d-1. If we keep on increasing the value of n, the remainders are cyclical in nature. The pattern of the remainders would repeat. Let us understand the concept of repetition with the help of an example.

Example 1: 4^1 divided by 9, leaves a remainder of 4.

4^2 divided by 9, leaves a remainder of 7. {Rem(16/9) = 7}

4^3 divided by 9, leaves a remainder of 1. {Rem (64/9) = 1}

4^4 divided by 9, leaves a remainder of 4. {Rem (256/9) = 4}

4^5 divided by 9, leaves a remainder of 4. {Rem (1024/9) = 7}

4^6 divided by 9, leaves a remainder of 4. {Rem (4096/9) = 1}

4^(3k+1) leaves a remainder of 4

4^(3k+2) leaves a remainder of 7

4^3k leaves a remainder of 1

As you can see above, the remainder when 4n is divided by 9 is cyclical in nature. The remainders obtained are 4, 7, 1, 4, 7, 1, 4, 7, 1 and so on. They will always follow the same pattern.

Concept

a^n when divided by d, will always give remainders which will have a pattern and will move in cycles of r such that r is less than or equal to d.