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Joined: 02 Jan 2020
Status:GRE Quant Tutor
Posts: 1111
Given Kudos: 9
Location: India
Concentration: General Management
Schools: XLRI Jamshedpur, India - Class of 2014
GMAT 1: 700 Q51 V31
GPA: 2.8
WE:Engineering (Computer Software)
How to Solve: Units' Digit of Power of 8
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22 Dec 2023, 04:46
22 Dec 2023, 04:46
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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 \(1757^{8979}\) ?
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: ::
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
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: ::
Sol: 57 divided by 4 gives 1 remainder
=> Units' digit of \(8^{57}\) = Units' digit of \(8^1\) = 8
=> Units' digit of \(8^{57}\) = Units' digit of \(8^1\) = 8
Q3. Find Units’ digit of \(8^{88}\)?
Show: ::
Sol: 88 divided by 4 gives 0 remainder
=> Units' digit of \(8^{88}\) = Units' digit of \(8^4\) = 6
=> Units' digit of \(8^{88}\) = Units' digit of \(8^4\) = 6
Q4. Find Units’ digit of \(8^{40a + 41}\) (given that a is a positive integer)?
Show: ::
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
= 0 + 1 = 1
=> Units' digit of \(8^{40a + 41}\) = Units' digit of \(8^1\) = 8
Q5. Find Units’ digit of \(1758^{8979}\)?
Show: ::
Sol: Units' digit of power of any number = Units' digit of power of the units' digit of that number
=> Units’ digit of \(1758^{8979}\) = 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 \(1758^{8979}\) = Units' digit of \(8^3\) = 2
=> Units’ digit of \(1758^{8979}\) = 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 \(1758^{8979}\) = Units' digit of \(8^3\) = 2
Master List of Units' Digit of Power of Numbers from 2 to 9
Hope it helps!
Re: How to Solve: Units' Digit of Power of 8
[#permalink]
26 Aug 2024, 07:42
26 Aug 2024, 07:42
- For Q3, I think the answer provided is wrong. If Remainder is 0 then units' digit will be 6, not 1.
Unit Digit of \(8^4 = 6\)
- The answer for Q5 doesn't make any sense.
Moderator
Joined: 02 Jan 2020
Status:GRE Quant Tutor
Posts: 1111
Given Kudos: 9
Location: India
Concentration: General Management
Schools: XLRI Jamshedpur, India - Class of 2014
GMAT 1: 700 Q51 V31
GPA: 2.8
WE:Engineering (Computer Software)
How to Solve: Units' Digit of Power of 8
[#permalink]
26 Aug 2024, 20:30
26 Aug 2024, 20:30
1
Thank you for reading the post and sharing the comment.
You are correct, there was a typo. Unit's digit of \(8^4\) = 6 as mentioned in the pattern in the post above.
Post has been corrected to reflect the same. Thank you.
Yes, the question is 1758^8979
Post has been corrected to reflect the same. Thank you.
dipenshome wrote:
- For Q3, I think the answer provided is wrong. If Remainder is 0 then units' digit will be 6, not 1.
Unit Digit of \(8^4 = 6\)
You are correct, there was a typo. Unit's digit of \(8^4\) = 6 as mentioned in the pattern in the post above.
Post has been corrected to reflect the same. Thank you.
dipenshome wrote:
- The answer for Q5 doesn't make any sense.
Yes, the question is 1758^8979
Post has been corrected to reflect the same. Thank you.
