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Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
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Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
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Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
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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 \(4^{51}\)?

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


Q2. Find Units’ digit of \(4^{33}\)?

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


Q3. Find Units’ digit of \(4^{44}\)?

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)?

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


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

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!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Expert Reply
Top Contributor
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 \(5^{61}\)?

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


Q2. Find Units’ digit of \(5^{33}\)?

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)?

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


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

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!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
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 \(6^{71}\)?

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


Q2. Find Units’ digit of \(6^{54}\)?

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)?

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


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

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!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Expert Reply
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 \(7^{81}\)?

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 \(7^{37}\)?

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


Q3. Find Units’ digit of \(7^{52}\)?

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)?

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 \(1297^{2041}\)?

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!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
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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 \(8^{81}\)?

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 \(8^{57}\)?

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


Q3. Find Units’ digit of \(8^{88}\)?

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)?

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 \(1738^{8979}\)?

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!
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [1]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
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Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
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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 \(9^{81}\)?

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


Q2. Find Units’ digit of \(9^{53}\)?

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


Q3. Find Units’ digit of \(9^{68}\)?

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)?

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


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

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
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
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