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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
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WE:Engineering (Computer Software)
How to Solve: Units' Digit of Power of 7
[#permalink]
21 Dec 2023, 04:10
21 Dec 2023, 04:10
Attachment:
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
[you-tube]https://www.youtube.com/watch?v=phDwwXPqHpo[/you-tube]
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: ::
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
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: ::
Sol: 37 divided by 4 gives 1 remainder
=> Units' digit of \(7^{37}\) = Units' digit of \(7^1\) = 7
=> Units' digit of \(7^{37}\) = Units' digit of \(7^1\) = 7
Q3. Find Units’ digit of \(7^{52}\)?
Show: ::
Sol: 52 divided by 4 gives 0 remainder
=> Units' digit of \(7^{52}\) = Units' digit of \(7^4\) = 1
=> 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: ::
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
= 0 + 3 = 3
=> Units' digit of \(7^{80a + 51}\) = Units' digit of \(7^3\) = 3
Q5. Find Units’ digit of \(1297^{2041}\)?
Show: ::
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
=> 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
Master List of Units' Digit of Power of Numbers from 2 to 9
Hope it helps!
gmatclubot
How to Solve: Units' Digit of Power of 7 [#permalink]
21 Dec 2023, 04:10
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