In most of the quantitative aptitude tests you might have faced questions like what’s the units digit of 9290 or 23512 etc. Needless to say, it’s practically impossible to do the direct calculation to find the answer considering the time constraint of quantitative aptitude exams. But if you notice, you can clearly see that there exists a pattern when you apply same mathematical operation continuously on a number. This is called cyclicity of numbers. In this paper we will have a look at top Cyclicity of Numbers aptitude questions, concepts and shortcuts on how to solve units digit problems quickly using concepts of number cyclicity
Problems based on cyclicity of numbers are common in most MBA entrance examinations like CAT, MAT, XAT, GMAT and other competitive exams like BANK tests. Questions related to number cyclicity have also started appearing in career recruitment tests or placement tests conducted by various companies. Since last few years TCS has redesigned their quantitative aptitude test patterns which include questions based on cyclicity of numbers. Questions like 'find the units digit in the result of ab ' are difficult to solve if you do them manually.
Instead if you apply theory of cyclicity of numbers, these questions does not take much of your time to solve them. Remember time is a constraint all competitive exams. So picking up concepts of cyclicity of numbers is definitely a bonus. Take the online test provided at the end to check your understanding level once you complete this tutorial on number cyclicity.
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Important Concepts and Shortcuts of Cyclicity of Numbers
There are mainly 3 categories of questions which fall under cyclicity of numbers, that include
- How to find units digit of ab
- How to find units digit of ab * cd * ef
- How to find units digit of abc
Find units digit of ab
Consider powers of 2
As we know,
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128.. and so on
Let's check the validity of above formula with an example.
Example
- Find the units place digit of 299?
Shortcuts to solve problems related to units place digit of ab
- Case 1: If b is a multiple of 4
- If a is an even number, ie: 2, 4, 6 or 8 then the units place digit is 6
- If a is an odd number, ie: 1, 3, 7 or 9 then the units place digit is 1
- Case 2: If b is not a multiple of 4
- Let r be the reminder when b is divided by 4, then units place of ab will be equal to units place of ar
Here we have captured the cyclicity of numbers upto 9 in the below table.
| Number | ^1 | ^2 | ^3 | ^4 | Cyclicity |
|---|---|---|---|---|---|
| 2 | 2 | 4 | 8 | 6 | 4 |
| 3 | 3 | 9 | 7 | 1 | 4 |
| 4 | 4 | 6 | 4 | 6 | 2 |
| 5 | 5 | 5 | 5 | 5 | 1 |
| 6 | 6 | 6 | 6 | 6 | 1 |
| 7 | 7 | 9 | 3 | 1 | 4 |
| 8 | 8 | 4 | 2 | 6 | 4 |
| 9 | 9 | 1 | 9 | 1 | 2 |
Find units digit for numbers of the form ab * cd
- First find the unit digit of ab and cd separately. Let the answers be x and y
- Then unit digit of ab * cd = units digit of x * y
Find units digit of abc
- Case 1: If cyclicity of units place digit of a is 4 then we have to divide the exponent of a by 4 and find out the remainder. Depending on the value of remainder we can apply the general rule of cyclicity given above and reach the solution.
- Case 2: If cyclicity of units place digit of a is 2, only extra information we need to find is if the exponent will be even or odd. Then we can apply the general rule of cyclicity given above and reach the solution.
Example 1:
- Find the units place digit of 24344
- Here cyclicity of units place digit is 4 (Units place digit is 2, from the above table we can see the cyclicity of 2 is 4). Hence case 1 is applicable.
- Now we have to find the remainder when exponent of 2 is divided by 4, that is the remainder when 4344 is divided by 4.
- Remainder of 4344/4 = Remainder of (44 – 1)44/4
- Using the binomial theorem, (as explained in number system tutorial) we can see that there is only one term in the expansion of (44 – 1)44 which is not divisible by 4.
- The term is 144/4
- Remainder of 144/4 = 1
- Now we can apply the general rules of cyclicity, (since reminder is 1, case 2 of general rule of cyclicity is applicable) which says, units place of 24344 = units place of 21 = 2.
- Find the units place digit of 293945
- Here cyclicity of units place digit of a, that is cyclicity of 9 is 2. Hence case 2 is applicable
- Next step is we need to find if 3945 is even or odd
- 3945 will always result in an odd number, because we are multiplying an odd number 39, odd number of times (45).
- Now we can apply the general rule of cyclicity described in the first section. General rule says, first check if exponent is divisible by 4. Since the exponent here is odd, it's not divisible by 4.
- Again according to general rule, if exponent is not divisible by 4, find the reminder when exponent is divided by 4.
- Since the exponent is odd here, the possible reminders when it's divided by 4 are 1 and 3.
- This means, the units place digit of number will be 1st or 3rd element in the cyclicity of units place digit of a.
In this case 1st and 3rd elements of 9 (units place digit of 29) is 9.
Hence units place digit of 2293945 is 9.
