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How to Solve: Units' Digit of Numbers by BrushMyQuant

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How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   Updated on: 02 Jan 2024, 07:28
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GMAT Quant Theory Master for All BrushMyQuant Units' Digit of Numbers Articles
(PDFs can be found in individual links given as GmatClub Article Link)



Units' Digit of Product of Numbers and Exponents




Units' Digit of Power of 2 and 3




Units' Digit of Power of 4 and 5




Units' Digit of Power of 6 and 7




Units' Digit of Power of 8 and 9


Attachment:
Cyclicity.jpg
Cyclicity.jpg [ 127.71 KiB | Viewed 357 times ]


Hope it helps!

Originally posted by BrushMyQuant on 23 Dec 2023, 06:56.
Last edited by BrushMyQuant on 02 Jan 2024, 07:28, edited 4 times in total.
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 06:57
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Attachment:
Units Digit of Power of 2.pdf [184.79 KiB]
Downloaded 26 times


How to Solve: Units' Digit of Power of 2


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 2



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 2
    ⁍ Find Units’ digit of \(2^{31}\) ?
    ⁍ Find Units’ digit of \(2^{93}\) ?
    ⁍ Find Units’ digit of \(2^{48}\) ?
    ⁍ Find Units’ digit of \(2^{20x + 74}\) (given that x is a positive integer)?
    ⁍ Find Units’ digit of \(132^{1955}\) ?


Theory of Units' Digit of Power of 2

• To find units' digit of any positive integer power of 2


We need to find the cycle of units' digit of power of 2
\(2^1\) units’ digit is 2
\(2^2\) units’ digit is 4
\(2^3\) units’ digit is 8
\(2^4\) units’ digit is 6		\(2^5\) units’ digit is 2
\(2^6\) units’ digit is 4
\(2^7\) units’ digit is 8
\(2^8\) units’ digit is 6


=> The power repeats after every \(4^{th}\) power
=> Cycle of units' digit of power of 2 = 4
=> We need to divide the power by 4 and check the remainder
=> Units' digit will be same as Units' digit of \(2^{Remainder}\)

NOTE: If Remainder is 0 then units' digit = units' digit of \(2^{Cycle}\) = units' digit of \(2^{4}\) = 6

Q1. Find Units’ digit of \(2^{31}\)?

Show Spoiler
Sol: We need to divided the power (31) by 4 and get the remainder
31 divided by 4 gives 3 remainder
=> Units' digit of \(2^{31}\) = Units' digit of \(2^3\) = 8


Q2. Find Units’ digit of \(2^{93}\)?

Show Spoiler
Sol: 93 divided by 4 gives 1 remainder
=> Units' digit of \(2^{93}\) = Units' digit of \(2^1\) = 2


Q3. Find Units’ digit of \(2^{48}\)?

Show Spoiler
Sol: 48 divided by 4 gives 0 remainder
=> Units' digit of \(2^{48}\) = Units' digit of \(2^4\) = 6


Q4. Find Units’ digit of \(2^{20x + 74}\) (given that x is a positive integer)?

Show Spoiler
Sol: Remainder of 20x + 74 divided by 4 = Remainder of 20x by 4 + Remainder of 74 by 4
= 0 + 2 = 2
=> Units' digit of \(2^{20x + 74}\) = Units' digit of \(2^2\) = 4


Q5. Find Units’ digit of \(132^{1955}\)?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(132^{1955}\) = Units’ digit of \(2^{1955}\)
=> Remainder of 1955 divided by 4 = Remainder of last two digits by 4 [ Watch this video to Master Divisibility Rules ]
=> Remainder of 55 by 4 = 3
=> Units' digit of \(132^{1955}\) = Units' digit of \(2^3\) = 8


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 06:57
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Attachment:
Units Digit of Power of 3.pdf [185.59 KiB]
Downloaded 14 times


How to Solve: Units' Digit of Power of 3


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 3



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 3
    ⁍ Find Units’ digit of \(3^{41}\) ?
    ⁍ Find Units’ digit of \(3^{97}\) ?
    ⁍ Find Units’ digit of \(3^{52}\) ?
    ⁍ Find Units’ digit of \(3^{40a + 71}\) (given that a is a positive integer)?
    ⁍ Find Units’ digit of \(133^{1855}\) ?


Theory of Units' Digit of Power of 3

• To find units' digit of any positive integer power of 3


We need to find the cycle of units' digit of power of 3
\(3^1\) units’ digit is 3
\(3^2\) units’ digit is 9
\(3^3\) units’ digit is 7
\(3^4\) units’ digit is 1		\(3^5\) units’ digit is 3
\(3^6\) units’ digit is 9
\(3^7\) units’ digit is 7
\(3^8\) units’ digit is 1


=> The power repeats after every \(4^{th}\) power
=> Cycle of units' digit of power of 3 = 4
=> We need to divide the power by 4 and check the remainder
=> Units' digit will be same as Units' digit of \(3^{Remainder}\)

NOTE: If Remainder is 0 then units' digit = units' digit of \(3^{Cycle}\) = units' digit of \(3^{4}\) = 1

Q1. Find Units’ digit of \(3^{41}\)?

Show Spoiler
Sol: We need to divided the power (41) by 4 and get the remainder
41 divided by 4 gives 1 remainder
=> Units' digit of \(3^{41}\) = Units' digit of \(3^1\) = 3


Q2. Find Units’ digit of \(3^{97}\)?

Show Spoiler
Sol: 97 divided by 4 gives 1 remainder
=> Units' digit of \(3^{97}\) = Units' digit of \(3^1\) = 3


Q3. Find Units’ digit of \(3^{52}\)?

Show Spoiler
Sol: 52 divided by 4 gives 0 remainder
=> Units' digit of \(3^{52}\) = Units' digit of \(3^4\) = 1


Q4. Find Units’ digit of \(3^{40a + 71 }\) (given that a is a positive integer)?

Show Spoiler
Sol: Remainder of 40a + 71 divided by 4 = Remainder of 40a by 4 + Remainder of 71 by 4
= 0 + 3 = 3
=> Units' digit of \(3^{40a + 71}\) = Units' digit of \(3^3\) = 7


Q5. Find Units’ digit of \(133^{1855}\)?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(133^{1855}\) = Units’ digit of \(3^{1855}\)
=> Remainder of 1855 divided by 4 = Remainder of last two digits by 4

Watch this video to Master Divisibility Rules

=> Remainder of 55 by 4 = 3
=> Units' digit of \(133^{1855}\) = Units' digit of \(3^3\) = 7


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 06:58
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Attachment:
Units Digit of Power of 4.pdf [158.34 KiB]
Downloaded 9 times


How to Solve: Units' Digit of Power of 4


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 4



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 4
    ⁍ Find Units’ digit of \(4^{51}\) ?
    ⁍ Find Units’ digit of \(4^{33}\) ?
    ⁍ Find Units’ digit of \(4^{44}\) ?
    ⁍ Find Units’ digit of \(4^{60x + 61}\) (given that x is a positive integer)?
    ⁍ Find Units’ digit of \(12954^{1053}\) ?


Theory of Units' Digit of Power of 4

• To find units' digit of any positive integer power of 4


We need to find the cycle of units' digit of power of 4
\(4^1\) units’ digit is 4
\(4^2\) units’ digit is 6		\(4^3\) units’ digit is 4
\(4^4\) units’ digit is 6


=> The power repeats after every \(2^{nd}\) power
=> Cycle of units' digit of power of 4 = 2

=> Units' digit of odd power of 4 = 4
=> Units' digit of even power of 4 = 6

Q1. Find Units’ digit of \(4^{51}\)?

Show Spoiler
Sol: 51 is Odd
=> Units' digit of \(4^{51}\) = 4


Q2. Find Units’ digit of \(4^{33}\)?

Show Spoiler
Sol: 33 is Odd
=> Units' digit of \(4^{33}\) = 4


Q3. Find Units’ digit of \(4^{44}\)?

Show Spoiler
Sol: 44 is Even
=> Units' digit of \(4^{Even}\) = 6


Q4. Find Units’ digit of \(4^{60x + 61}\) (given that x is a positive integer)?

Show Spoiler
Sol: 60x + 61 = Even + Odd = Odd
=> Units' digit of \(4^{60x + 61}\) = 4


Q5. Find Units’ digit of \(12954^{1053}\) ?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(12954^{1053}\) = Units’ digit of \(4^{1053}\)
=> 1053 is Odd
=> Units' digit of \(12954^{1053}\) = Units, digit of \(4^{1053}\)= 4


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 06:59
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Attachment:
Units Digit of Power of 5.pdf [162.84 KiB]
Downloaded 25 times


How to Solve: Units' Digit of Power of 5


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 5



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 5
    ⁍ Find Units’ digit of \(5^{61}\) ?
    ⁍ Find Units’ digit of \(5^{33}\) ?
    ⁍ Find Units’ digit of \(5^{79x + 31}\) (given that x is a positive integer)?
    ⁍ Find Units’ digit of \(1055^{199}\) ?


Theory of Units' Digit of Power of 5

• To find units' digit of any positive integer power of 5


We need to find the cycle of units' digit of power of 5
\(5^1\) units’ digit is 5\(5^2\) units’ digit is 5


=> Units’ digit of any positive integer power of 5 = 5

Q1. Find Units’ digit of \(5^{61}\)?

Show Spoiler
Sol: Since 61 is a positive integer
=> Units' digit of \(5^{61}\) = 5


Q2. Find Units’ digit of \(5^{33}\)?

Show Spoiler
Sol: Since 33 is a positive integer
=> Units' digit of \(5^{33}\) = 5


Q3. Find Units’ digit of \(5^{79x + 31}\) (given that x is a positive integer)?

Show Spoiler
Sol: Since 79x + 31 is a positive integer
=> Units' digit of \(5^{79x + 31}\) = 5


Q4. Find Units’ digit of \(1055^{199}\) ?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(1055^{199}\) = Units’ digit of \(5^{199}\)
Since 199 is a positive integer
=> Units' digit of \(1055^{199}\) = 5


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 06:59
Expert Reply
Top Contributor
Attachment:
Units Digit of Power of 6.pdf [162.88 KiB]
Downloaded 14 times


How to Solve: Units' Digit of Power of 6


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 6



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 6
    ⁍ Find Units’ digit of \(6^{71}\) ?
    ⁍ Find Units’ digit of \(6^{54}\) ?
    ⁍ Find Units’ digit of \(6^{70x + 41}\) (given that x is a positive integer)?
    ⁍ Find Units’ digit of \(2756^{205}\) ?


Theory of Units' Digit of Power of 6

• To find units' digit of any positive integer power of 6


We need to find the cycle of units' digit of power of 6
\(6^1\) units’ digit is 6\(6^2\) units’ digit is 6


=> Units’ digit of any positive integer power of 6 = 6

Q1. Find Units’ digit of \(6^{71}\)?

Show Spoiler
Sol: Since 71 is a positive integer
=> Units' digit of \(6^{71}\) = 6


Q2. Find Units’ digit of \(6^{54}\)?

Show Spoiler
Sol: Since 54 is a positive integer
=> Units' digit of \(6^{54}\) = 6


Q3. Find Units’ digit of \(6^{70x + 41}\) (given that x is a positive integer)?

Show Spoiler
Sol: Since 70x + 41 is a positive integer
=> Units' digit of \(6^{70x + 41}\) = 6


Q4. Find Units’ digit of \(2756^{205}\) ?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(2756^{205}\) = Units’ digit of \(6^{205}\)
Since 205 is a positive integer
=> Units' digit of \(2756^{205}\) = 6


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 07:00
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Top Contributor
Attachment:
Units Digit of Power of 7.pdf [185.38 KiB]
Downloaded 17 times


How to Solve: Units' Digit of Power of 7


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 7



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 7
    ⁍ Find Units’ digit of \(7^{81}\) ?
    ⁍ Find Units’ digit of \(7^{37}\) ?
    ⁍ Find Units’ digit of \(7^{52}\) ?
    ⁍ Find Units’ digit of \(7^{80a + 51}\) (given that a is a positive integer)?
    ⁍ Find Units’ digit of \(1297^{2041}\) ?


Theory of Units' Digit of Power of 7

• To find units' digit of any positive integer power of 7


We need to find the cycle of units' digit of power of 7
\(7^1\) units’ digit is 7
\(7^2\) units’ digit is 9
\(7^3\) units’ digit is 3
\(7^4\) units’ digit is 1		\(7^5\) units’ digit is 7
\(7^6\) units’ digit is 9
\(7^7\) units’ digit is 3
\(7^8\) units’ digit is 1


=> The power repeats after every \(4^{th}\) power
=> Cycle of units' digit of power of 7 = 4
=> We need to divide the power by 4 and check the remainder
=> Units' digit will be same as Units' digit of \(7^{Remainder}\)

NOTE: If Remainder is 0 then units' digit = units' digit of \(7^{Cycle}\) = units' digit of \(7^{4}\) = 1

Q1. Find Units’ digit of \(7^{81}\)?

Show Spoiler
Sol: We need to divided the power (81) by 4 and get the remainder
81 divided by 4 gives 1 remainder
=> Units' digit of \(7^{81}\) = Units' digit of \(7^1\) = 7


Q2. Find Units’ digit of \(7^{37}\)?

Show Spoiler
Sol: 37 divided by 4 gives 1 remainder
=> Units' digit of \(7^{37}\) = Units' digit of \(7^1\) = 7


Q3. Find Units’ digit of \(7^{52}\)?

Show Spoiler
Sol: 52 divided by 4 gives 0 remainder
=> Units' digit of \(7^{52}\) = Units' digit of \(7^4\) = 1


Q4. Find Units’ digit of \(7^{80a + 51 }\) (given that a is a positive integer)?

Show Spoiler
Sol: Remainder of 80a + 51 divided by 4 = Remainder of 80a by 4 + Remainder of 51 by 4
= 0 + 3 = 3
=> Units' digit of \(7^{80a + 51}\) = Units' digit of \(7^3\) = 3


Q5. Find Units’ digit of \(1297^{2041}\)?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(1297^{2041}\) = Units’ digit of \(7^{2041}\)
=> Remainder of 2041 divided by 4 = Remainder of last two digits by 4

Watch this video to Master Divisibility Rules

=> Remainder of 41 by 4 = 1
=> Units' digit of \(1297^{2041}\) = Units' digit of \(7^1\) = 7


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 07:00
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Top Contributor
Attachment:
Units Digit of Power of 8.pdf [185.75 KiB]
Downloaded 17 times


How to Solve: Units' Digit of Power of 8


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 8



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 8
    ⁍ Find Units’ digit of \(8^{91}\) ?
    ⁍ Find Units’ digit of \(8^{57}\) ?
    ⁍ Find Units’ digit of \(8^{88}\) ?
    ⁍ Find Units’ digit of \(8^{40a + 41}\) (given that a is a positive integer)?
    ⁍ Find Units’ digit of \(1738^{8979}\) ?


Theory of Units' Digit of Power of 8

• To find units' digit of any positive integer power of 8


We need to find the cycle of units' digit of power of 8
\(8^1\) units’ digit is 8
\(8^2\) units’ digit is 4
\(8^3\) units’ digit is 2
\(8^4\) units’ digit is 6		\(8^5\) units’ digit is 8
\(8^6\) units’ digit is 4
\(8^7\) units’ digit is 2
\(8^8\) units’ digit is 6


=> The power repeats after every \(4^{th}\) power
=> Cycle of units' digit of power of 8 = 4
=> We need to divide the power by 4 and check the remainder
=> Units' digit will be same as Units' digit of \(8^{Remainder}\)

NOTE: If Remainder is 0 then units' digit = units' digit of \(8^{Cycle}\) = units' digit of \(8^{4}\) = 1

Q1. Find Units’ digit of \(8^{81}\)?

Show Spoiler
Sol: We need to divided the power (81) by 4 and get the remainder
81 divided by 4 gives 1 remainder
=> Units' digit of \(8^{81}\) = Units' digit of \(8^1\) = 8


Q2. Find Units’ digit of \(8^{57}\)?

Show Spoiler
Sol: 57 divided by 4 gives 1 remainder
=> Units' digit of \(8^{57}\) = Units' digit of \(8^1\) = 8


Q3. Find Units’ digit of \(8^{88}\)?

Show Spoiler
Sol: 88 divided by 4 gives 0 remainder
=> Units' digit of \(8^{88}\) = Units' digit of \(8^4\) = 1


Q4. Find Units’ digit of \(8^{40a + 41}\) (given that a is a positive integer)?

Show Spoiler
Sol: Remainder of 40a + 41 divided by 4 = Remainder of 40a by 4 + Remainder of 41 by 4
= 0 + 1 = 1
=> Units' digit of \(8^{40a + 41}\) = Units' digit of \(8^1\) = 8


Q5. Find Units’ digit of \(1738^{8979}\)?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(1738^{8979}\) = Units’ digit of \(8^{8979}\)
=> Remainder of 8979 divided by 4 = Remainder of last two digits by 4

Watch this video to Master Divisibility Rules

=> Remainder of 79 by 4 = 3
=> Units' digit of \(1738^{8979}\) = Units' digit of \(8^3\) = 2


Hope it helps!
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink] New post   23 Dec 2023, 07:01
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Kudos
Expert Reply
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Attachment:
Units Digit of Power of 9.pdf [158.67 KiB]
Downloaded 13 times


How to Solve: Units' Digit of Power of 9


Hi All,

I have posted a video on YouTube to discuss Units' Digit of Power of 9



Attached pdf of this Article as SPOILER at the top! Happy learning! :)

Following is Covered in the Video

Theory of Units' Digit of Power of 9
    ⁍ Find Units’ digit of \(9^{81}\) ?
    ⁍ Find Units’ digit of \(9^{53}\) ?
    ⁍ Find Units’ digit of \(9^{68}\) ?
    ⁍ Find Units’ digit of \(9^{60x + 61}\) (given that x is a positive integer)?
    ⁍ Find Units’ digit of \(13259^{1279}\) ?


Theory of Units' Digit of Power of 9

• To find units' digit of any positive integer power of 9


We need to find the cycle of units' digit of power of 9
\(9^1\) units’ digit is 9
\(9^2\) units’ digit is 1		\(9^3\) units’ digit is 9
\(9^9\) units’ digit is 1


=> The power repeats after every \(2^{nd}\) power
=> Cycle of units' digit of power of 9 = 2

=> Units' digit of odd power of 9 = 9
=> Units' digit of even power of 9 = 1

Q1. Find Units’ digit of \(9^{81}\)?

Show Spoiler
Sol: 81 is Odd
=> Units' digit of \(9^{81}\) = 9


Q2. Find Units’ digit of \(9^{53}\)?

Show Spoiler
Sol: 53 is Odd
=> Units' digit of \(9^{53}\) = 9


Q3. Find Units’ digit of \(9^{68}\)?

Show Spoiler
Sol: 68 is Even
=> Units' digit of \(9^{Even}\) = 1


Q9. Find Units’ digit of \(9^{60x + 61}\) (given that x is a positive integer)?

Show Spoiler
Sol: 60x + 61 = Even + Odd = Odd
=> Units' digit of \(9^{60x + 61}\) = 9


Q5. Find Units’ digit of \(13259^{1279}\) ?

Show Spoiler
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(13259^{1279}\) = Units’ digit of \(9^{1279}\)
=> 1279 is Odd
=> Units' digit of \(13259^{1279}\) = Units, digit of \(9^{1279}\)= 9


Hope it helps!
GMAT Club Bot
gmatclubot
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
23 Dec 2023, 07:01
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