Hi all,
I was searching around and could not find this information easily, so took a few minutes to throw it together to share in case anyone else needed it.
Units Digit Shortcut / Patterns
1 = [1]
2 = [2,4,8,6]
3 = [3,9,7,1]
4 = [4,6]
5 = [5]
6 = [6]
7 = [7,9,3,1]
8 = [8,4,2,6]
9 = [9,1]
0 = [0]
Some are easy to memorize, such as 1, 4, 5, 6, 9, and 0. The others can be quickly figured out by going through a few powers of the number until it repeats.
For example, let's take a number ending in 3. (I'll use ANEI to mean "a number ending in").
ANEI 3 * ANEI 3 = ANEI 9 (Example: 3*3=9, or 3*13=39)
ANEI 9 * ANEI 3 = ANEI 7 (Example: 9*3=27, or 9*13=117)
ANEI 7 * ANEI 3 = ANEI 1 (Example: 7*3=21, or 17*3=51)
ANEI 1 * ANEI 3 = ANEI 3 (REPEAT!)
So the pattern for ANEI 3 will be [3,9,7,1].
This means that ANEI 3^1,5,9,13,(+4 etc.) will have a Units Digit of 3.
ANEI 3^2,6,10,14,(+4 etc.) will have a Units Digit of 9.
ANEI 3^3,7,11,15,(+4 etc.) will have a Units Digit of 7.
and ANEI 3^4,8,12,16,(+4 etc.) will have a Units Digit of 1.
Example: What is the units digit of 768493023^31?
3 has a pattern containing 4 numbers.
31/4 = 7 r3
The third item in the pattern is 7, thus 768493023^31 will have a units digit of 7.
Another example, this time from the Manhattan GMAT Advanced Gmat Quant book:
For 17^3, the pattern is [7,9,3,1], so the units digit is 3 (since it is the third in the pattern).
24^5+2x isn't as tricky as it seems.
If x=1, then 5+2(1) = 7 (odd)
If x=2, then 5+2(2) = 9 (odd).
So regardless of the value of x, the power is odd.
ANEI 4^(odd power) = ANEI 4. This is because the pattern for 4 is [4 (odd), 6 (even)].
So when we multiply ANEI 6 * ANEI 3 * ANEI 4 = ANEI 8 * ANEI 4 = ANEI 2.
The answer is A.
Hope this comes in handy on test day!
Oh, and if anyone has any information on how to apply this to tens digits or hundreds digits, please share!
Thanks,
--Rishi
I was searching around and could not find this information easily, so took a few minutes to throw it together to share in case anyone else needed it.
Units Digit Shortcut / Patterns
1 = [1]
2 = [2,4,8,6]
3 = [3,9,7,1]
4 = [4,6]
5 = [5]
6 = [6]
7 = [7,9,3,1]
8 = [8,4,2,6]
9 = [9,1]
0 = [0]
Some are easy to memorize, such as 1, 4, 5, 6, 9, and 0. The others can be quickly figured out by going through a few powers of the number until it repeats.
For example, let's take a number ending in 3. (I'll use ANEI to mean "a number ending in").
ANEI 3 * ANEI 3 = ANEI 9 (Example: 3*3=9, or 3*13=39)
ANEI 9 * ANEI 3 = ANEI 7 (Example: 9*3=27, or 9*13=117)
ANEI 7 * ANEI 3 = ANEI 1 (Example: 7*3=21, or 17*3=51)
ANEI 1 * ANEI 3 = ANEI 3 (REPEAT!)
So the pattern for ANEI 3 will be [3,9,7,1].
This means that ANEI 3^1,5,9,13,(+4 etc.) will have a Units Digit of 3.
ANEI 3^2,6,10,14,(+4 etc.) will have a Units Digit of 9.
ANEI 3^3,7,11,15,(+4 etc.) will have a Units Digit of 7.
and ANEI 3^4,8,12,16,(+4 etc.) will have a Units Digit of 1.
Example: What is the units digit of 768493023^31?
3 has a pattern containing 4 numbers.
31/4 = 7 r3
The third item in the pattern is 7, thus 768493023^31 will have a units digit of 7.
Another example, this time from the Manhattan GMAT Advanced Gmat Quant book:
We can easily find the units digit of 36^6, since ANEI 6 * ANEI 6 = ANEI 6.If x is a positive integer, what is the units digit of (24)^5+2x (36)^6 (17)^3?
A. 2
B. 3
C. 4
D. 6
E. 8
For 17^3, the pattern is [7,9,3,1], so the units digit is 3 (since it is the third in the pattern).
24^5+2x isn't as tricky as it seems.
If x=1, then 5+2(1) = 7 (odd)
If x=2, then 5+2(2) = 9 (odd).
So regardless of the value of x, the power is odd.
ANEI 4^(odd power) = ANEI 4. This is because the pattern for 4 is [4 (odd), 6 (even)].
So when we multiply ANEI 6 * ANEI 3 * ANEI 4 = ANEI 8 * ANEI 4 = ANEI 2.
The answer is A.
Hope this comes in handy on test day!
Oh, and if anyone has any information on how to apply this to tens digits or hundreds digits, please share!
Thanks,
--Rishi
