Number System has its significance in almost every topic in Mathematics. It is enough to describe its importance. Having a sound knowledge in this topic, helps the aspirant in getting the command over other topics of APTITUDE.
Although almost all the topics of NUMBER SYSTEM are important. However, UNIT DIGIT holds a special place in it. Because in all of the exams, the chances of appearing the questions from this topic are comparatively higher.
So it becomes the duty of us to provide you a profound knowledge of this area. Minding the relevance of the topic we, www.gyaanshalaa.blogspot.com is here to serve you by providing you the complete concepts and questions that would give you a wide glance about the Chapter.
The concept of unit digit works upon the cyclicity of numbers which means mainly about the last digit of the number and its repetitive pattern on being multiplied by the same number.
To explain the concept of UNIT DIGIT, we bifurcated the method on the basis of being EVEN and ODD UNIT DIGIT.
EVEN UNIT DIGIT NUMBERS : 2, 4, 6, 8, 0
Explanation for the Cyclicity of 2 :
We can clearly see that after every 4 terms, the unit digit of the “2 raised to the power” repeats .
Hence we can say that the cyclicity of 2 is 4
Similarly we can also calculate the cyclicity of other EVEN NUMBERS.
The cyclicity of all the EVEN NUMBERS turns out to be :
While calculating the CYCLICITY of EVEN NUMBER, we come across that :
2⁴ = 6
4⁴ = 6
6⁴ = 6
8⁴ = 6
Therefore, We can easily generalize that
(EVEN NUMBERS)⁴ⁿ = 6
Here n = 1, 2, 3, 4, ………..
ODD UNIT DIGIT = 3, 7, 9
Explanation for the Cyclicity of 3 :
We can clearly see that after every 4 terms, the unit digit of the “3 raised to the power” repeats .
Hence we can say that the cyclicity of 3 is 4
Similarly we can also calculate the cyclicity of other ODD NUMBERS.
The cyclicity of all the ODD NUMBERS turns out to be :
While calculating the CYCLICITY of ODD NUMBER, we come across that :
3⁴ = 1
7⁴ = 1
9⁴ = 1
Therefore, We can easily generalise that
(ODD NUMBERS)⁴ⁿ = 1
Here n = 1, 2, 3, 4, ………..
EXCEPTIONS :
There are certain terms of which CYCLICITY is 1, that means the value repeats itself when raised to any power.
Such digits are : 0, 1, 5 and 6
It means
0 ^ n = 0
1^ n = 1
5 ^ n = 5
6 ^ n = 6
CYCLICITY TABLE :
Summarised form of the cyclicity concept discussed above :
FOOD FOR THOUGHT :
(EVEN NUMBERS)⁴ⁿ = 6
(ODD NUMBERS)⁴ⁿ = 1
0 ^ n = 0
1^ n = 1
5 ^ n = 5
6 ^ n = 6
Let’s apply these concepts and hands on questions to have proficiency over the topic.
PRACTICE QUESTIONS :
Q 1. Find the unit digit of :
7²⁹⁵
72⁴⁵⁶
249⁶⁵
84⁸⁷
91³²¹⁴
Sol : a. We know that ; last digit is odd number i.e. 7
(ODD NUMBERS)⁴ⁿ = 1
7²⁹⁵ = 7²⁹² x 7³ = 1 x 3 = 3
Unit Digit of 7²⁹⁵ is 3.
b. We know that ; last digit is even number i.e. 2
(EVEN NUMBERS)⁴ⁿ = 6
72⁴⁵⁶ = 6
Unit digit of 72⁴⁵⁶ is 6.
c. We know that ; last digit is odd number i.e. 9
(ODD NUMBERS)⁴ⁿ = 1
249⁶⁵ = 249⁶⁴ x 249¹ = 1 c 9 = 9
Unit digit of 249⁶⁵ is 9.
d. We know that ; last digit is even number i.e. 4
(EVEN NUMBERS)⁴ⁿ = 6
84⁸⁷ = 84⁸⁴ x 84 ³ = 6 x 4 = 4
Unit digit of 84⁸⁷ is 4.
e. We know that ; last digit is 1
1^ n = 1
91³²¹⁴ = 1
Unit digit of 91³²¹⁴ is 1.
Q 2. The digit in the unit place of the number 9²⁸⁹ x 2⁶⁷ is
4
7
2
6
Sol : Unit digit of 9²⁸⁹ = 9²⁸⁸ x 9 ¹ = 1 x 9 = 9
Unit digit of 2⁶⁷ = 2⁶⁴ x 2³ = 6 x 8 = 8
Unit place of the number 9²⁸⁹ x 2⁶⁷ = 9 x 8 = 2
Q3. Find the unit digit of 12345 x 7896 ?
6
5
0
None of these
Sol : To calculate the unit digit, we are supposed to use the unit digit of the numbers
Therefore, 5 x 6 = 30
Hence, the unit digit of 12345 x 7896 is 0.
Q4. Find the unit digit of 147⁴⁵⁶⁷³¹ + 562³²⁵⁶⁷⁸ ?
3
9
7
6
Sol: Unit digit of 147⁴⁵⁶⁷³¹ = 1 x 3 = 3
Unit digit of 562³²⁵⁶⁷⁸ = 6 x 4 = 4
Therefore, 3 + 4 = 7
Hence, the unit digit of 147⁴⁵⁶⁷³¹ + 562³²⁵⁶⁷⁸ is 7.
Q 5. Find the unit digit of 147⁴⁵⁶⁷⁵ - 562³²⁵⁶⁷⁹ ?
3
5
6
9
Sol : Unit digit of 147⁴⁵⁶⁷⁵ = 1 x 3 = 3
Unit digit of 562³²⁵⁶⁷⁹ = 6 x 8 = 8
Therefore, 3 - 8 = 5
Hence the unit digit of 147⁴⁵⁶⁷⁵ - 562³²⁵⁶⁷⁹ is 5.
Q 6. What is the unit digit of 2! + 4! + 6! + ……….+ 98! ?
0
4
6
2
Sol : 2! = 2 x 1 = 2
4! = 4 x 3 x 2 x 1 = 24
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
.
.
.
98! = 98 x 96 x 94 x ……………..x 3 x 2 x 1 = Value ends with Zero
Hence, the unit digit of 2! + 4! + 6! + ……….+ 98! = 2 + 24 + 720 + ….. = Unit digit will be 6.
Q 7. What is the unit digit of 13^(14^(15^(16^17))) ?
2
1
9
7
Sol : 16¹⁷ = Unit digit → 6
15⁶ = Unit digit → 5
14⁵ = Unit digit → 4
13⁴ = Unit digit → 1
Hence , the unit digit of 13^(14^(15^(16^17))) is 1.
Q 8. What is the unit digit of 1! + 2! + 3! + ……….+ 100! ?
4
3
7
9
Sol : 1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Similarly, 5! To 100 ! all have 0 at their Unit Place.
Hence, the unit digit of 1! + 2! + 3! + ……….+ 100!
= 1 + 2 + 6 + 24 + 120 + ……
= Unit Digit → 3.
Q 9. Which of the following is the unit digit of 3¹⁹⁸⁶ - 2¹⁹⁸⁶ ?
4
5
6
9
Sol: Unit digit of 3¹⁹⁸⁶ = 3¹⁹⁸⁴ x 3² = 1 x 9 = 9
Unit digit of 2¹⁹⁸⁶ = 2¹⁹⁸⁴ x 2² = 6 x 4 = 4
Hence, the unit digit of 3¹⁹⁸⁶ - 2¹⁹⁸⁶ = 9 - 4 = 5
Q 10. What is the unit digit of 1 + 9¹ + 9² + 9³ + …….. 9¹⁰⁰⁰⁶ ?
0
1
7
9
Sol : It can be written as :
9⁰ + 9¹ + 9² + 9³ + …….. 9¹⁰⁰⁰⁶
Unit digit of 9⁰ = 1
Unit digit of 9¹ = 9
Unit digit of 9² = 1
Unit digit of 9³ = 9
On observing the pattern, we get that
Unit digit of 9 raised to the power ^ EVEN = 1
Unit digit of 9 raised to the power ^ ODD = 1
Sum of all the even power + odd powers → 10 → Unit Digit → 0
Only 9¹⁰⁰⁰⁶ = Unit digit of 9¹⁰⁰⁰⁶ = 1
Hence , the unit digit of 1 + 9¹ + 9² + 9³ + …….. 9¹⁰⁰⁰⁶ = 1
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