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Updated on: 27 Sep 2024, 05:02
Originally posted by BrushMyQuant on 23 Dec 2023, 06:56.
Last edited by BrushMyQuant on 27 Sep 2024, 05:02, edited 6 times in total.
Last edited by BrushMyQuant on 27 Sep 2024, 05:02, edited 6 times in total.
Kudos
Bookmarks
GMAT Quant Theory Master for All BrushMyQuant Units' Digit of Numbers Articles
(PDFs can be found in individual links given as GmatClub Article Link)
Cyclicity.jpg [ 127.71 KiB | Viewed 5740 times ]
Link to Theory for Last Two digits of exponents here.
Hope it helps!
(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 [ 127.71 KiB | Viewed 5740 times ]
Link to Theory for Last Two digits of exponents here.
Hope it helps!
23 Dec 2023, 06:57
Kudos
Bookmarks
Attachment:
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 \(2^{31}\)?
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
31 divided by 4 gives 3 remainder
=> Units' digit of \(2^{31}\) = Units' digit of \(2^3\) = 8
Q2. Find Units’ digit of \(2^{93}\)?
Sol: 93 divided by 4 gives 1 remainder
=> Units' digit of \(2^{93}\) = Units' digit of \(2^1\) = 2
=> Units' digit of \(2^{93}\) = Units' digit of \(2^1\) = 2
Q3. Find Units’ digit of \(2^{48}\)?
Sol: 48 divided by 4 gives 0 remainder
=> Units' digit of \(2^{48}\) = Units' digit of \(2^4\) = 6
=> 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)?
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
= 0 + 2 = 2
=> Units' digit of \(2^{20x + 74}\) = Units' digit of \(2^2\) = 4
Q5. Find Units’ digit of \(132^{1955}\)?
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
=> 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!
23 Dec 2023, 06:57
Kudos
Bookmarks
Attachment:
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 \(3^{41}\)?
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
41 divided by 4 gives 1 remainder
=> Units' digit of \(3^{41}\) = Units' digit of \(3^1\) = 3
Q2. Find Units’ digit of \(3^{97}\)?
Sol: 97 divided by 4 gives 1 remainder
=> Units' digit of \(3^{97}\) = Units' digit of \(3^1\) = 3
=> Units' digit of \(3^{97}\) = Units' digit of \(3^1\) = 3
Q3. Find Units’ digit of \(3^{52}\)?
Sol: 52 divided by 4 gives 0 remainder
=> Units' digit of \(3^{52}\) = Units' digit of \(3^4\) = 1
=> 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)?
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
= 0 + 3 = 3
=> Units' digit of \(3^{40a + 71}\) = Units' digit of \(3^3\) = 7
Q5. Find Units’ digit of \(133^{1855}\)?
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
=> 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!
