Cyclicity & Remainders (for CAT)

Number Cyclicity:-

Number cyclicity is very important for every chapter of the quant. Any exams will definitely need the basics of number cyclicity. Questions can directly be asked on this topic and you will need this to solve many questions on other topics.

Don’t forget time is biggest constrain for any entrance exam and cyclicity will have one to save time.

Cyclicity is basically use to find the unit digit or tens digit of the number.

Unit Digit Cyclicity:-

Q1. Find the unit digit of 2^2548.

Sol: - You will need more than 6 hours to solve this problem if you don’t use cyclicity theorem.

We notice that

2^1 end with 2

2^2 end with 4

2^3 end with 8

2^4 end with 6

2^5 end with 2

2^6 end with 4

2^7 end with 8

2^8 end with 6

We notice that 5^th power end in 2 and number repeats after 4 powers. Hence cyclicity for 2 is 4. It will always end with 2, 4, 6 and 8.

So Divide 2548 by 4 and we get remainder = 2

Hence unit digit of 2^2548 with be 4.

Remember:-

When exponent is 1 number ends with 2

When exponent is 2 number ends with 4

When exponent is 3 number ends with 8

When exponent is 0 or 4 number ends with 6.

Similarly we can find of all other numbers.

Number	Unit digit of Cyclicity
0,1,5,6	1
4,9	2
2,3,7,8	4

Tens Digit Cyclicity:-

Similarly we can arrive for tens digit cyclicity.

Number Tens Digit Cyclicity

Number	Unit digit of Cyclicity
0,1,5,6	1
7	4
6	5
4,9	10
2,3,8	20