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# How to Solve: Units' Digit of Numbers by BrushMyQuant

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Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1777
Own Kudos [?]: 2094 [2]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
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)
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)
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)
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Top Contributor
Attachment:
Units Digit of Power of 4.pdf [158.34 KiB]

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)
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Top Contributor
Attachment:
Units Digit of Power of 5.pdf [162.84 KiB]

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)
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Top Contributor
Attachment:
Units Digit of Power of 6.pdf [162.88 KiB]

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)
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Top Contributor
Attachment:
Units Digit of Power of 7.pdf [185.38 KiB]

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)
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
Top Contributor
Attachment:
Units Digit of Power of 8.pdf [185.75 KiB]

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)
Re: How to Solve: Units' Digit of Numbers by BrushMyQuant [#permalink]
1
Kudos
Top Contributor
Attachment:
Units Digit of Power of 9.pdf [158.67 KiB]

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